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<img src="/qimages/44101" /><p>COMPARING FUNCTIONS A linear function models the cost of renting a truck from a moving company. The table shows the cost <code class='latex inline'>\displaystyle y </code> (in dollars) when you drive the truck <code class='latex inline'>\displaystyle x </code> miles. Graph the function and compare the slope and the <code class='latex inline'>\displaystyle y </code> -intercept of the graph with the slope and the <code class='latex inline'>\displaystyle c </code> -intercept of the graph in Exercise 38 . Miles, <code class='latex inline'>\displaystyle \boldsymbol{x} </code> & Cost (dollars), <code class='latex inline'>\displaystyle \boldsymbol{y} </code> 0 & 40 50 & 80 100 & 120</p>

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<img src="/qimages/43316" />

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<p>Chapter Problem Your first task as producer of Canadian Superstar is to rent a theatre for the first event, a singing competition. Rental includes lunch and snacks for the competitors. Details for the two best choices are shown:</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|}
\hline Royal james Hall \$50 per person \\
\hline
\end{array}
</code> <code class='latex inline'>\displaystyle
\begin{array}{|l|}
\hline Broadway Nights \\
\$ 1000 plus \$ 30 per person \\
\hline
\end{array}
</code> 10</p><p>You have <code class='latex inline'>\displaystyle \$ 2000 </code> in your budget for this event. You would like to begin the competition with as many contestants as you can afford.</p><p>The cost for renting Broadway Nights can be modelled by the equation <code class='latex inline'>\displaystyle C=1000+30 n </code>. Explain why this equation correctly gives the cost, <code class='latex inline'>\displaystyle C </code>, in dollars, for <code class='latex inline'>\displaystyle n </code> contestants.</p>

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Learning Path

L1
Quick Intro to Factoring Trinomial with Leading a

L2
Introduction to Factoring ax^2+bx+c

L3
Factoring ax^2+bx+c, ex1

Now You Try

<p>Aaron did his homework at school with a graphing calculator. He determined that the equation of the line of best fit for some data was <code class='latex inline'>y=2.63x-1.29</code>. Once he got home, he realized he had mixed up the independent and dependent variables. Write the correct equation for the relation in the form <code class='latex inline'>y=mx+b</code>.</p>

<p>Chapter Problem Your first task as producer of Canadian Superstar is to rent a theatre for the first event, a singing competition. Rental includes lunch and snacks for the competitors. Details for the two best choices are shown:</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|}
\hline Royal james Hall \$50 per person \\
\hline
\end{array}
</code> <code class='latex inline'>\displaystyle
\begin{array}{|l|}
\hline Broadway Nights \\
\$ 1000 plus \$ 30 per person \\
\hline
\end{array}
</code> 10</p><p>You have <code class='latex inline'>\displaystyle \$ 2000 </code> in your budget for this event. You would like to begin the competition with as many contestants as you can afford.</p><p>Use the total amount budgeted <code class='latex inline'>\displaystyle (C=2000) </code> to solve this equation for <code class='latex inline'>\displaystyle n </code>.</p>

<p>A rental agency charges $8 per hour to rent a canoe.</p><p>a) Describe the relationship between the cost of the canoe rental and the
time, in hours, the canoe is rented for.</p><p>b) Illustrate the relationship graphically and represent it with an equation.</p><p>c) Use your graph to estimate the cost of renting the canoe for 8 h.</p><p>d) Use your equation to determine the exact cost of renting the canoe for 8 h.</p>

<p>Identify the constant of variation.</p><p><code class='latex inline'>\displaystyle 4 y-5 x=0 </code></p>

<p>Identify the constant of variation.</p><p><code class='latex inline'>\displaystyle y=\frac{3}{2} x </code></p>

<img src="/qimages/43369" /><p>MATHEMATICAL CONNECTIONS Rewrite each</p><p>geometry formula using function notation. Evaluate each function when <code class='latex inline'>\displaystyle r=5 </code> feet. Then explain the meaning of the result.</p><p>a. Diameter, <code class='latex inline'>\displaystyle d=2 r </code></p><p>b. Area, <code class='latex inline'>\displaystyle A=\pi r^{2} </code></p><p>c. Circumference, <code class='latex inline'>\displaystyle C=2 \pi r </code></p>

<p>Graph each function. The domain is the set of real numbers. Find the range.</p><p><code class='latex inline'>\displaystyle y=3 x-4 </code></p>

<p>The total cost of potatoes varies directly with the mass, in kilograms, bought. Potatoes cost $2.18/kg.</p><p>a) Choose appropriate letters for variables. Make a table of values showing the cost of 0 kg, 1 kg, 2 kg, 3 kg, 4 kg, and 5 kg of potatoes.</p><p>b) Graph the relationship.</p><p>c) Write an equation for the relationship in the form <code class='latex inline'>y = kx</code>.</p>

<p>For each function, determine whether y varies directly with x. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hline x & y \\ \hline 27 & 9 \\ \hline 30 & 10 \\ \hline 60 & 20 \\ \hline\end{array} </code></p>

<p>Judy is training for an Ironman triathlon race. During her training program, she finds that she can swim at 1.5 km/h, cycle at 30 km/h, and run at 12 km/h. To estimate her time for an upcoming race, Judy rearranges the formula did <code class='latex inline'>distance = speed\times time</code> to find that <code class='latex inline'>time = \dfrac{distance}{speed}</code>.</p><p>a) Choose a variable to represent the distance travelled for each part of the race. For example, choose 6 for cycle.</p><p>b) Copy and complete the table. The second row is done for you.</p><img src="/qimages/22208" /><p>c) Write a trinomial to model Judy’s time.</p><p>d) The upcoming Ironman race is a triathlon composed of a 3.8-km swim, a 180.2-km cycle, and a full marathon run of 42.2 km. Using your expression from part c), calculate how long it will take Judy to finish the race.</p>

<p>Kwok is a hotel manager. His responsibilities include renting rooms
for conferences. The hotel charges $250 per day plus $15 per person for the grand ballroom.</p><p>How many people could attend a wedding reception if the
wedding planners have a budget of</p>
<ul>
<li><code class='latex inline'>\$4000</code>?</li>
<li><code class='latex inline'>\$2000</code>?</li>
</ul>

<p>To raise money for a local charity, students organized a walk-a-thon, For the walk-a-thon, the amount of money raised by each student varies directly with the number of kilometres walked, Dieter raised $320 by walking 20 km.</p><p>a) Graph this direct variation for distances from 0 km to 20 km, using pencil and paper or technology,</p><p>b) Write an equation relating the money Dieter raised and the distance, in kilometres, that he walked.</p><p>c) How much would he have raised by walking 25 km?</p>

<p> A cell phone company charges a monthly fee of $30, plus $0.02 per minute of call time.</p><p>a) Write the monthly cost function, <code class='latex inline'>C(t)</code>, where t is the amount of time in minutes of call time during a month.</p><p>b) Find the domain and range of <code class='latex inline'>C</code>.</p>

<p>The table shows the cost, <code class='latex inline'>C</code>, in dollars, to rent a car for a day and drive a distance, <code class='latex inline'>d</code>, in kiliometres.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline Distance, \boldsymbol{d}(\mathbf{k m}) & Cost, C (\$) \\
\hline 0 & 50 \\
\hline 100 & 65 \\
\hline 200 & 80 \\
\hline 300 & 95 \\
\hline 400 & 110 \\
\hline
\end{array}
</code></p><p>a) What is the fixed cost?</p><p>b) What is the variable cost? Explain how you found this.</p><p>c) Write an equation relating to C and d.</p><p>d) What is the cost of renting a car for a day and driving 750 km?</p>

<p>Kwok is a hotel manager. His responsibilities include renting rooms
for conferences. The hotel charges $250 per day plus $15 per person for the grand ballroom.</p><p>How much should Kwok charge to rent the hall for</p>
<ul>
<li> 50 people?</li>
<li>100 people?the ballroom to the number of people, <code class='latex inline'>n</code>.</li>
</ul>

<p>Mr. Martinez is a sales representative for an agricultural supply company. He receives a salary and monthly commission. He also receives a bonus each time he reaches a sales goal.</p><p>Suppose Mr. Martinez’s annual salary is $42,000 and his average commission is $825 each month. If he receives four bonuses of $750 each, how much does he earn in a year?</p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y=4 x-3 </code></p>

<p>Justine’s mother is building an ultralight airplane. The fuel tank is made of plastic and has a mass of 5000 g. Each litre of gasoline has a mass of 840 g. The total mass of the fuel plus the tank cannot exceed 21 800 g.</p><p>The equation that represents this situation is: <code class='latex inline'>5000+840n=218000</code>.</p><p> Solve the equation to determine the number of litres of fuel in a full tank.</p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y=4 x+1 </code></p>

<img src="/qimages/43247" /><p>HOW DO YOU SEE IT? The graph represents the height <code class='latex inline'>\displaystyle h </code> of a projectile after <code class='latex inline'>\displaystyle t </code> seconds.</p><p>a. Explain why <code class='latex inline'>\displaystyle h </code> is a function of <code class='latex inline'>\displaystyle t </code>.</p><p>b. Approximate the height of the projectile after <code class='latex inline'>\displaystyle 0.5 </code> second and after <code class='latex inline'>\displaystyle 1.25 </code> seconds.</p><p>c. Approximate the domain of the function.</p><p>d. Is <code class='latex inline'>\displaystyle t </code> a function of <code class='latex inline'>\displaystyle h </code> ? Explain.</p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y=6 x </code></p>

<p>Make a table of <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-values and use it to graph the direct variation equation.</p><p><code class='latex inline'>\displaystyle y=-9 x </code></p>

<p>The Home Medical Supplies Rental Company charges according to the equation <code class='latex inline'>60m-C+75=0</code> to rent hospital beds, where <code class='latex inline'>C</code> represents the cost, in dollars, which depends on <code class='latex inline'>m</code>, the number of months that the bed is rented for. </p><p>a) Express the equation in slope <code class='latex inline'>y-intercept</code> form: <code class='latex inline'>C=mn+b</code>.</p><p>b) Identify the fixed and variable costs. </p><p>c) Illustrate the relation graphically using pencil and paper or a graphing calculator. </p><p>d) What is the rental cost if a hospital bed is rented for 5 months?</p>

<p>Ashleigh can walk 2 m/s and swim 1 m/s. What is the quickest way for Ashleigh to get from one corner of her pool to the opposite corner?</p><img src="/qimages/10107" /><p>a) Predict whether it is faster for Ashleigh to walk or swim.</p><p>b) Ashleigh can walk at a speed of 2 m/s. The time, in seconds, for Ashleigh to walk is <code class='latex inline'>\frac{w}{2}</code>, where <code class='latex inline'>w</code> is the distance, in metres, she
walks. Use this relationship to find the travel time if Ashleigh walks around the pool.</p><p>Path 1: Walk the entire distance.</p><img src="/qimages/10108" /><p>c) Write a similar expression to represent the time taken for Ashleigh to swim a distance s. Her swimming speed is 1 m/s. Use this relationship to find the travel time if Ashleigh swims straight across.</p><p>Path 2: Swim the entire distance.</p><img src="/qimages/10109" /><p>d) Which route is faster, and by how much?</p>

<p>A company rents cars for <code class='latex inline'>\$50</code> per day plus <code class='latex inline'>\$0.15/km</code>.</p><p><strong>(a)</strong> Express the daily rental cost as a function of the number of kilometres travelled.</p><p><strong>(b)</strong> Determine the rental cost if you drive 472 km in one day.</p><p><strong>(c)</strong> Determine how far you can drive in a day for $80. </p>

<p>Create a linear function machine and two points that are generated by the machine. Trade points with a classmate to determine the function that generated the points.</p>

<img src="/qimages/12124" /><p>The table shows the population of bacteria in a petri dish at various times. If the pattern continues, what will the bacteria population be at <code class='latex inline'>\displaystyle 6: 00 </code> P.M.? <code class='latex inline'>\displaystyle \begin{array}{|l|c|c|c|c|}\hline { Bacteria Population } \\\\ \hline Time & 8: 00 A.M. & 10:00 A.M. & 12:00 P.M. & 2: 00 p.M. \\\\ \hline Population & 100 & 200 & 400 & 800 \\\\ \hline\end{array} </code></p>

<p>At the season finale, you present the winner of Canadian Superstar with a recording-and-tour contract. The contract states that the winer will be paid <code class='latex inline'>\$ 5000</code> per month while on tour plus <code class='latex inline'>\$2</code> per CD sold.</p><p>In Canada, a record album or CD achieves gold status once it sells 50 000 units. How much will artist make if the CD goes gold after 6 months of touring?</p>

<p>Distance For a given speed, the distance traveled varies directly with the time. Kate's school is 5 miles away from her home and it takes her 10 minutes to reach the school. If Josh lives 2 miles from school and travels at the same speed as Kate, how long will it take him to reach the school?</p>

<p>Determine whether the graph, table, or equation represents a linear or nonlinear function. Explain. (Section 3.2)</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline \boldsymbol{x} & \boldsymbol{y} \\
\hline -5 & 3 \\
0 & 7 \\
5 & 10 \\
\hline
\end{array}
</code></p>

<p>CCSS SENSE-MAKING The table shows the median home prices in the United States, from 2007 to <code class='latex inline'>\displaystyle 2009 . </code> <code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hline Year & Median Home Price (S) \\ \hline 2007 & 234,300 \\ \hline 2008 & 213,200 \\ \hline 2009 & 212,200 \\ \hline\end{array} </code></p>

<img src="/qimages/43366" /><p>PROBLEM SOLVING The function <code class='latex inline'>\displaystyle C(x)=25 x+50 </code> represents the & Hours & Cost \cline { 1 - 3 } labor cost (in dollars) for Certified & 2 & <code class='latex inline'>\displaystyle \$ 130 </code> Remodeling to build a deck, where & 4 & <code class='latex inline'>\displaystyle \$ 160 </code> <code class='latex inline'>\displaystyle x </code> is the number of hours of labor. & 6 & <code class='latex inline'>\displaystyle \$ 190 </code> The table shows sample labor costs & & from its main competitor, Master & & Remodeling. The deck is estimated to take 8 hours of labor. Which company would you hire? Explain.</p>

<p>The Gala Restaurant uses the equation <code class='latex inline'>30n-C+200=0</code> to determine the cost for a room rental, where C represents the cost, in dollars, which depends on <code class='latex inline'>n</code>, the number of people attending. </p><p>a) Express the equation in slope <code class='latex inline'>y-intercept</code> form: <code class='latex inline'> C=mn+b</code>.</p><p>b) Identify the fixed and variable costs. </p><p>c) Illustrate the relation graphically using pencil and paper or a graphing calculator.</p><p>d) What is the rental cost if 100 people attend a hockey banquet. </p>

<p>The Tent-All Company charges according to the equation <code class='latex inline'>10d-C+50=0</code> to rent tents for camping, where <code class='latex inline'>C</code> represents the cost, in dollars, which depends on <code class='latex inline'>d</code>, the days that the tent is rented for. </p><p>a) Express the equation in slope <code class='latex inline'>y-intercept</code> form: <code class='latex inline'>C=mn+b </code>.</p><p>b) Identify the fixed and variable costs. </p><p>c) Illustrate the relation graphically. </p><p>d) What is the rental cost if a tent is rented for 7 days. </p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (-3,-7) </code></p>

<p>Erin joins a CD club. The first 10 CDs are free, but after that she pays <code class='latex inline'>\$15.95</code> for each CD she orders.</p><p>It can be modelled by <code class='latex inline'>
\displaystyle
Cost = 15.95x - 159.5
</code></p>
<ul>
<li>Erin receives her first order of CDs with a bill for <code class='latex inline'>\$31.90</code>. Create and solve an equation to determine how many she ordered.</li>
</ul>

<p>A taxi company charges a fare of $2.25 plus $0.75 per mile traveled. The cost of the fare <code class='latex inline'>c</code> can be described by the equation <code class='latex inline'>c=0.75m+2.25</code>, where <code class='latex inline'>m</code> is the number of miles traveled.</p><p>If you need to travel 18 miles, how much will the taxi fare cost?</p>

<p>Reasoning Explain why you cannot answer the following question. If <code class='latex inline'>\displaystyle y=0 </code> when <code class='latex inline'>\displaystyle x=0 </code>, what is <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=13 ? </code></p>

<p>Think About a Plan In a factory, a certain machine needs 10 min to warm up. It takes 15 min for the machine to run a cycle. The machine can operate for as long as <code class='latex inline'>\displaystyle 6 \mathrm{~h} </code> per day including warm-up time. Draw a graph showing the total time the machine operates during 1 day as a function of the number of cycles it runs.</p>
<ul>
<li><p>What domain and range are reasonable?</p></li>
<li><p>Is the function a linear function?</p></li>
</ul>

<p>STATE FAIR The Ohio State Fair charges <code class='latex inline'>\displaystyle \$ 8 </code> for admission and <code class='latex inline'>\displaystyle \$ 5 </code> for parking. After Joey pays for admission and parking, he plans to spend all of his remaining money at the ring game, which cósts <code class='latex inline'>\displaystyle \$ 3 </code> per game.</p><p>a. Write an equation representing the situation.</p><p>b. How much did Joey spend at the fair if he paid <code class='latex inline'>\displaystyle \$ 6 </code> for food and drinks and played the ring game 4 times?</p>

<p>You earn <code class='latex inline'>\displaystyle \$ 10 </code> for each hour you work at a canoe rental shop. Write an expression for your salary for working the number of hours <code class='latex inline'>\displaystyle h </code>. Make a table to find how much you earn for working <code class='latex inline'>\displaystyle 10 \mathrm{~h}, 20 \mathrm{~h}, 30 \mathrm{~h} </code>, and <code class='latex inline'>\displaystyle 40 \mathrm{~h} </code>.</p>

<p>For each function, determine whether y varies directly with x. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hline x & y \\ \hline 2 & 14 \\ \hline 3 & 21 \\ \hline 5 & 35 \\ \hline\end{array} </code></p>

<img src="/qimages/43246" />
<ol>
<li>ANALYZING RELATIONSHIPS You select items in a vending machine by pressing one letter and then one number.</li>
</ol>
<p>a. Explain why the relation that pairs letter-number combinations with food or drink items is a function.</p><p>b. Identify the independent and dependent variables.</p><p>c. Find the domain and range of the function.</p>

<p>Without graphing, tell whether the slope of a line that models each linear relationship is positive, negative, zero, or undefined. Then find the slope.</p><p>A student earns a 98 on a test for answering one question incorrectly and earns a 90 for answering five questions incorrectly.</p>

<p>Identify the slope and the vertical intercept of each linear relation and explain what they represent. Write an equation to describe the relationship.</p><img src="/qimages/22078" />

<p>You can use the formula <code class='latex inline'>C =2.5I</code> to obtain an approximate value for converting a length, <code class='latex inline'>I</code>, in inches to a length, <code class='latex inline'>C</code>, in centimetres.</p><p>Use the formula to find the number of centimetres in</p>
<ul>
<li>6 inches</li>
<li>3 feet (1 foot = 12 inches)</li>
</ul>

<p>Describe and correct the error in the statement about the relation shown in the table. </p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|l|l|l|l|l|}
\hline Input, \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\
\hline Output, \boldsymbol{y} & 6 & 7 & 8 & 6 & 9 \\
\hline
\end{array}
</code></p><p>The relation is a function. The</p><p>range is <code class='latex inline'>\displaystyle 1,2,3,4 </code>, and <code class='latex inline'>\displaystyle 5 . </code></p>

<p>The Everything for Events Rental Company charges according to the equation <code class='latex inline'>25n-C+100=0</code> to rent tables for events, where C represents the cost, in dollars, which depends on <code class='latex inline'>n</code>, the number of tables that are rented. </p><p>a) Express the equations in slope <code class='latex inline'>y-intercept</code> form: <code class='latex inline'>C=mn+b</code></p><p>b) Identify the fixed and variable costs. </p><p>c) Illustrate the relation graphically using pencil and paper or a graphing calculator. </p><p>d) What is the rental cost if 200 tables are rented for a charity event?</p>

<p>A telephone company charges $30 a month and gives the customer 200 free call minutes. After the 200 min, the company charges $0.03 a minute.</p><p>a) Write the function using function notation.</p><p>b) Find the cost for talking 350 min in a month.</p><p>c) Find the cost for talking 180 min in a month.</p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y-6 x=0 </code></p>

<p>Determine the constant of variation for the direct variation.</p><p>The distance traveled by a car varies directly with time. The car travels 270 km in 3 h.</p>

<img src="/qimages/44101" /><p>COMPARING FUNCTIONS A linear function models the cost of renting a truck from a moving company. The table shows the cost <code class='latex inline'>\displaystyle y </code> (in dollars) when you drive the truck <code class='latex inline'>\displaystyle x </code> miles. Graph the function and compare the slope and the <code class='latex inline'>\displaystyle y </code> -intercept of the graph with the slope and the <code class='latex inline'>\displaystyle c </code> -intercept of the graph in Exercise 38 . Miles, <code class='latex inline'>\displaystyle \boldsymbol{x} </code> & Cost (dollars), <code class='latex inline'>\displaystyle \boldsymbol{y} </code> 0 & 40 50 & 80 100 & 120</p>

<p>Error Analysis Identify the error in the statement shown at the right.</p><p>If y varies directly with <code class='latex inline'>\displaystyle x^{2} </code>, and <code class='latex inline'>\displaystyle y=2 </code> when <code class='latex inline'>\displaystyle x=4 </code>, then <code class='latex inline'>\displaystyle y=3 </code> when <code class='latex inline'>\displaystyle x=9 </code>.</p>

<p>The volume of water in a water tank varies with time. The tank contains
200 L of water after 2 min.</p><p>a) Write an equation relating the volume of water and time. What does the constant of variation represent?</p><p>b) Graph this relationship using pencil and paper or technology.</p><p>c) What volume of water is in the tank after 30 min?</p><p>d) How long will it take to fill a water tank that can hold 100 000 L of
water?</p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y+3=-3 x </code></p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (-3,14) </code></p>

<p>Skylar wants to join the local gym.
The cost in dollars for a membership can be expressed as:
<code class='latex inline'>100 + 39.99m</code></p><p>where 100 is the initiation fee in dollars, 39.99 is the monthly fee in dollars, and m is the number of months for which a person signs up.</p><p>How much will it cost Skylar to join the gym for 14 months?</p>

<p>Identify the slope and the vertical intercept of each linear relation and explain what they represent. Write an equation to describe the relationship.</p><img src="/qimages/22079" />

<p>Chapter Problem Your first task as producer of Canadian Superstar is to rent a theatre for the first event, a singing competition. Rental includes lunch and snacks for the competitors. Details for the two best choices are shown:</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|}
\hline Royal james Hall \$50 per person \\
\hline
\end{array}
</code> <code class='latex inline'>\displaystyle
\begin{array}{|l|}
\hline Broadway Nights \\
\$ 1000 plus \$ 30 per person \\
\hline
\end{array}
</code> 10</p><p>You have <code class='latex inline'>\displaystyle \$ 2000 </code> in your budget for this event. You would like to begin the competition with as many contestants as you can afford.</p>
<ul>
<li>Which hall should you rent? Explain.</li>
</ul>

<p>Chapter Problem Your first task as producer of Canadian Superstar is to rent a theatre for the first event, a singing competition. Rental includes lunch and snacks for the competitors. Details for the two best choices are shown:</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|}
\hline Royal james Hall \$50 per person \\
\hline
\end{array}
</code> <code class='latex inline'>\displaystyle
\begin{array}{|l|}
\hline Broadway Nights \\
\$ 1000 plus \$ 30 per person \\
\hline
\end{array}
</code> 10</p><p>You have <code class='latex inline'>\displaystyle \$ 2000 </code> in your budget for this event. You would like to begin the competition with as many contestants as you can afford.</p>
<ul>
<li>Write an equation to model the cost for renting Royal James Hall. Solve the equation.</li>
</ul>

<p>Charlene earns <code class='latex inline'>\displaystyle \$ 150 </code> more per week than her roommate Kristi and <code class='latex inline'>\displaystyle \$ 100 </code> less than her other roommate, Sacha. Together the three friends earn <code class='latex inline'>\displaystyle \$ 2050 </code> per week. How much does each girl earn per week?</p>

<p>To convert from Canadian (= British Imperial) gallons to litres, multiply 4.546. Write an equation to convert litres to Canadian gallons. Round the constant of variation to the nearest thousandth.</p>

<p>Mr. Martinez is a sales representative for an agricultural supply company. He receives a salary and monthly commission. He also receives a bonus each time he reaches a sales goal.</p><p>Let e represent earnings, s represent his salary, c represent his commission, and b represent his bonus. Write an algebraic expression to represent his earnings if he receives four equal bonuses.</p>

<p>Under water, pressure increases 4.3 pounds per square inch (psi) for every 10 feet you descend. This can be expressed by the equation <code class='latex inline'>p=0.43d+14.7</code>, where <code class='latex inline'>p</code> is the pressure in pounds per square inch and <code class='latex inline'>d</code> is the depth in feet.</p><p>Divers cannot work at depths below about 400 feet. What is the pressure at this depth?</p>

<p>Identity each relation as a direct variation. a partial variation, or neither.</p><p>Justify your answer.</p><p>a) <code class='latex inline'>y = 10x</code></p><p>b) <code class='latex inline'>C = 4t +3</code></p><p>c) <code class='latex inline'>y = 3x + 2</code></p>

<p>Since pure gold is very soft, other metals are often added to it to make an alloy that is stronger and more durable. The relative amount of gold in a piece of jewelry is measured in karats. The formula for the relationship is <code class='latex inline'>g=\frac{25k}{6}</code>, where <code class='latex inline'>k</code> is the number of karats and <code class='latex inline'>g</code> is the percent of gold in the jewelry.</p><p>How many karats are in a ring that is 50% gold?</p>

<p>The function <code class='latex inline'>\displaystyle y=3.5 x+2.8 </code> represents the cost <code class='latex inline'>\displaystyle y </code> (in dollars) of a taxi ride of <code class='latex inline'>\displaystyle x </code> miles.</p><p>a. Identify the independent and dependent variables.</p><p>b. You have enough money to travel at most 20 miles in the taxi. Find the domain and range of the function.</p>

<p>The amount of blood in the body can be predicted by the equation <code class='latex inline'>y=0.07w</code>, where <code class='latex inline'>y</code> is the number of pints of blood and <code class='latex inline'>w</code> is the weight of a person in pounds.</p><p>Graph the equation.</p>

<p>RECREATION You want to make sure that you have enough music for a car trip. If each <code class='latex inline'>\displaystyle \mathrm{CD} </code> is an average of 45 minutes long, the linear function <code class='latex inline'>\displaystyle m(x)=0.75 x </code> could be used to find out how many CDs you need to bring.</p><p>a. How many hours of music are there on <code class='latex inline'>\displaystyle 4 \mathrm{CDs} </code> ?</p><p>b. If the trip you are taking is 6 hours, how many CDs should you bring?</p>

<p>Think About a Plan Suppose the equation <code class='latex inline'>\displaystyle y=12+10 x </code> models the amount of money in your wallet, where <code class='latex inline'>\displaystyle y </code> is the total in dollars and <code class='latex inline'>\displaystyle x </code> is the number of weeks from today. If you graphed this equation, what would the slope represent in the situation? Explain.</p>
<ul>
<li><p>Is the equation in slope-intercept form?</p></li>
<li><p>What units make sense for the slope?</p></li>
</ul>

<p>When treating a sick child, a doctor may need to estimate the child's body surface area (BSA). This information helps the doctor determine appropriate doses of medicine. The BSA, in square centimetres, for a child can be estimated using the formula <code class='latex inline'>\displaystyle \mathrm{BSA}=1321+0.3433 \mathrm{~m} </code>, where <code class='latex inline'>\displaystyle m </code> represents the</p><p>child's mass in kilograms.</p><p>a) Find the BSA for a child</p><p>with mass <code class='latex inline'>\displaystyle 12 \mathrm{~kg} </code>. b) Suppose that to receive a</p><p>certain treatment, a child</p><p>must have a BSA greater</p><p>than <code class='latex inline'>\displaystyle 1333 \mathrm{~cm}^{2} </code>. What is the minimum mass a child</p><p>must have, to receive this</p><p>treatment?</p>

<ol>
<li>Mach number An aircraft breaks the sound barrier when it flies at about <code class='latex inline'>\displaystyle 1200 \mathrm{~km} / \mathrm{h} </code>. This speed is known as Mach <code class='latex inline'>\displaystyle 1 . </code> The Mach number, <code class='latex inline'>\displaystyle M </code>, is given by the function <code class='latex inline'>\displaystyle M=\frac{s}{1200} </code>, where <code class='latex inline'>\displaystyle s </code> is the speed of the aircraft in kilometres per hour. a) What is the value of <code class='latex inline'>\displaystyle M </code> when <code class='latex inline'>\displaystyle s=2400 </code> ? when <code class='latex inline'>\displaystyle s=3000 </code> ? b) Communication In the function defined by the ordered pairs (speed, Mach number), identify the dependent variable and the independent variable. Explain your reasoning.</li>
</ol>

<p>The first three diagrams in a pattern are shown.</p><img src="/qimages/6536" /><p>a) Write a formula for the total number of small squares in the nth diagram.</p><p>b) Write a formula for the number of shaded small squares in the nth diagram.</p><p>c) Write a formula for the number of unshaded small squares in the nth diagram.</p><p>d) Write your formula from part c) in factored form.</p><p>e) Show that both forms of the formula give the same results for the 15th diagram.</p>

<p>Johannes Kepler (1571-1630) was a German astronomer who noticed a pattern in the orbits of planets. The table shows data for the planets known when Kepler was alive.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|c|}
\hline Planet & Radius of Orbit (AU) ^{\star} & Period of Orbit (Earth Days) \\
\hline Mercury & 0.389 & 87.77 \\
\hline Venus & 0.724 & 224.70 \\
\hline Earth & 1.0 & 365.25 \\
\hline Mars & 1.524 & 686.98 \\
\hline Jupiter & 5.200 & 4332.62 \\
\hline Saturn & 9.150 & 10759.20 \\
\hline
\end{array}
</code></p><p>Kepler conjectured that the square of the period divided by the cube of the radius is a constant.</p>
<ul>
<li><code class='latex inline'>\displaystyle A U </code>, or astronomical unit, is the mean distance from Earth to the Sun, <code class='latex inline'>\displaystyle 1.49 \times 10^{8} \mathrm{~km} </code>.</li>
</ul>
<p>Write a formula for the relationship that Kepler found. This is called Kepler's Third Law.</p>

<p>Banquet hall A uses the equation <code class='latex inline'>25n-C+1250=0</code> to
determine the cost for a hall rental.</p><p>Banquet Hall B uses the equation <code class='latex inline'>30n+995-C=0</code> to determine the cost for their hall rental.</p><p>In each case, <code class='latex inline'>C</code> represents the cost, in dollars, which depends on $n$, the number of people attending.</p><p>Express each equation in slope y-intercept form: <code class='latex inline'>C=mn+b</code>.</p><p>(a) Identify the fixed and variable costs for each hall.</p><p>(b) What is the cost at each hall for a graduation banquet for
45 people?</p><p>(c) Which hall offers the better price? Comment on whether your
conclusion changes if a few more people wish to attend.</p>

<p>Physics Light travels about <code class='latex inline'>\displaystyle 186,000 \mathrm{mi} / \mathrm{s} </code>. The function <code class='latex inline'>\displaystyle d(t)=186,000 t </code> gives the distance <code class='latex inline'>\displaystyle d(t) </code>, in miles, that light travels in <code class='latex inline'>\displaystyle t </code> seconds. How far does light travel in 30 s?</p>

<p>At the season finale, you present the winner of Canadian Superstar with a recording-and-tour contract. The contract states that the winer will be paid <code class='latex inline'>\$ 5000</code> per month while on tour plus <code class='latex inline'>\$2</code> per CD sold.</p><p>Write an equation that relates total earning in terms of the number of months, m, on tour and the number, n , of CD sold.</p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y=-5 x </code></p>

<p>Kyle sells used cars. He is paid <code class='latex inline'>\$14</code>/hour plus an <code class='latex inline'>8\%</code> commission on sales. What dollar amount of car sales must Kyle make to earn <code class='latex inline'>\$1200</code> in a 38-h work week?</p>

<p>Latisha is beginning an exercise program that calls for 20 minutes
of walking each day for the first week. Each week thereafter, she has to increase her walking by 7 minutes a day. Which week of her exercise program will be the first one in which she will walk over an hour a day?</p>

<p>A banquet hall charges according to the equation <code class='latex inline'>\displaystyle C=25 n+250 </code>, where <code class='latex inline'>\displaystyle C </code> represents the total cost in dollars to rent the hall, and <code class='latex inline'>\displaystyle n </code> represents the number of people attending the event. If the total cost to rent the hall for a particular event was <code class='latex inline'>\displaystyle \$ 375 </code>, how many people attended the event?</p>

<p>Kwan coaches baseball. She has <code class='latex inline'>\displaystyle \$ 450 </code> to buy uniforms for the team. Each uniform costs <code class='latex inline'>\displaystyle \$ 30 </code>. a) Write an equation showing the relationship between the total cost in dollars, <code class='latex inline'>\displaystyle C </code>, and the number of uniforms, <code class='latex inline'>\displaystyle n </code>. b) There are 16 players on the team. Will there be enough uniforms? Explain.</p>

<p><code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>.</p><p>If <code class='latex inline'>\displaystyle y=25 </code> when <code class='latex inline'>\displaystyle x=15 </code>, find <code class='latex inline'>\displaystyle y </code> when <code class='latex inline'>\displaystyle x=6 </code>.</p>

<img src="/qimages/44186" /><p>ANALYZING RELATIONSHIPS You have <code class='latex inline'>\displaystyle \$ 50 </code> to spend on fabric for a blanket. The amount <code class='latex inline'>\displaystyle m </code> (in dollars) of money you have after buying <code class='latex inline'>\displaystyle y </code> yards of fabric is given by the function <code class='latex inline'>\displaystyle m(y)=-9.98 y+50 </code>. How does the graph of <code class='latex inline'>\displaystyle m </code> change in each situation?</p><p>a. You receive an additional <code class='latex inline'>\displaystyle \$ 10 </code> to spend on the fabric.</p><p>b. The fabric goes on sale, and each yard now costs <code class='latex inline'>\displaystyle \$ 4.99 . </code></p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (2,-9) </code></p>

<p>A group of musicians who made a CD are paid according to the
following breakdown, where <code class='latex inline'>n</code> is the number of CDs sold.</p>
<ul>
<li>The table shows sales achievement levels for the Canadian
recording industry. </li>
</ul>
<img src="/qimages/1554" /><p>Find the total amount paid to the group if their CD.</p>
<ul>
<li>i) sells 100 copies</li>
<li>ii) reaches gold status</li>
<li>iii) reaches diamond status</li>
</ul>

<p>As a thunderstorm approaches, you see lightning as it occurs, but you hear the accompanying sound of thunder a short time afterward. The distance <code class='latex inline'>d</code> in miles that sound travels in <code class='latex inline'>t</code> seconds is given by the equation <code class='latex inline'>d=0.21t</code>.</p><p>Make a table of values.</p>

<p>Three authors team up to write a children’s book. The publisher pays them according to the following contracts.</p><img src="/qimages/1140" />
<ul>
<li>Find a simplified expression that represents the total that the publisher must pay the writing team.</li>
</ul>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle y=-2 x </code></p>

<p>Under water, pressure increases 4.3 pounds per square inch (psi) for every 10 feet you descend. This can be expressed by the equation <code class='latex inline'>p=0.43d+14.7</code>, where <code class='latex inline'>p</code> is the pressure in pounds per square inch and <code class='latex inline'>d</code> is the depth in feet.</p><p>How many times as great is the pressure at 400 feet as the pressure at sea level?</p>

<p>The sum of the first <code class='latex inline'>n</code> even natural numbers can be found using the formula <code class='latex inline'>S=n(n+1)</code></p><p>Verify the formula for <code class='latex inline'>n=1</code> and <code class='latex inline'>n=2</code>. </p>
<ul>
<li>What is the sum of the first five even natural numbers?</li>
</ul>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (7,2) </code></p>

<p>Determine whether y varies directly with x. If so, find the constant of variation.</p><p><code class='latex inline'>\displaystyle
y = 12x
</code></p>

<ol>
<li>CCSS MODELING Miguel is earning extra money by painting houses. He charges a <code class='latex inline'>\displaystyle \$ 200 </code> fee plus <code class='latex inline'>\displaystyle \$ 12 </code> per can of paint needed to complete the job. Write and use an equation to find how many cans of paint he needs for a <code class='latex inline'>\displaystyle \$ 260 </code> job.</li>
</ol>

<p>Determine whether <code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hlinex & y \\ \hline 23 & 24 \\ \hline 55 & 56 \\ \hline 66 & 67 \\ \hline\end{array} </code></p>

<p>Paloma works part-time, 4h per day, selling fitness club memberships. She is paid <code class='latex inline'>\$9/h</code>, plus a <code class='latex inline'>\$12</code> commission for each 1-year memberships she sells.</p><p> Paloma notices that her sales have a pattern: for the first 12 h of the week she sells an average of two memberships per hour and for the lats 12 h of the week she sells an average of three memberships per hour. Use an organized method (e.g. chart, graph, equation) to determine when Paloma will reach a special <code class='latex inline'>\$900</code> earning goal.</p>

<p>a) Copy and complete the table of values given that y varies partially with x.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline x & y \\
\hline 0 & 4 \\
\hline 1 & 7 \\
\hline 2 & \\
\hline 3 & 13 \\
\hline 4 & \\
\hline & 25 \\
\hline
\end{array}
</code></p><p>b) Identify the initial value of <code class='latex inline'>y</code> and the constant of variation from the table.</p><p>c) Write an equation relating y and x in the form <code class='latex inline'>y = mx + b</code>.</p><p>d) Graph the relation. Describe the graph.</p>

<p>The graph shows how the total cost, in dollars, to ship workbooks is related</p><p>to the number of workbooks.</p><p>a) Write an equation in the form</p><p><code class='latex inline'>\displaystyle y=m x+b </code> for this relation. b) What do the values of <code class='latex inline'>\displaystyle m </code> and <code class='latex inline'>\displaystyle b </code> represent?</p><p>c) Jee-Yun has a budget of <code class='latex inline'>\displaystyle \$ 200 </code> for workbooks. How many can she buy?</p><img src="/qimages/156494" />

<p>Which form would you use to write the equation of a line if you knew its slope and <code class='latex inline'>\displaystyle x </code>-intercept? Explain.</p>

<img src="/qimages/11025" /><p>The distance-time graph of a person</p><p>walking at a constant speed in front</p><p>of a motion sensor is shown.</p><p>a) How far from the motion sensor was</p><p>the person when she began walking?</p><p>b) Was she moving toward or away from</p><p>the sensor? Explain how you know.</p><p>c) How fast was she walking?</p><p>d) Write an equation that describes this distance-time relationship.</p>

<p>If <code class='latex inline'>\displaystyle x </code> is divided by 7, what happens to <code class='latex inline'>\displaystyle y ? </code></p>

<p>For each function, determine whether y varies directly with x. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{c|c}x & y \\ \hline 11 & 22 \\ 16 & 32 \\ 7 & 42 \\ \hline\end{array} </code></p>

<p>Use the table below that shows the maximum heart rate to maintain, for different ages, during aerobic activities such as running, biking, or swimming.</p><img src="/qimages/24684" /><p>What would be the maximum heart rate to maintain in aerobic training for a 10-year old? an 80-year old?</p>

<p>Paloma works part-time, 4h per day, selling fitness club memberships. She is paid <code class='latex inline'>\$9/h</code>, plus a <code class='latex inline'>\$12</code> commission for each 1-year memberships she sells.</p><p>Write an algebraic expression that describes Paloma's total earnings.</p>

<p>Murray works at a cell phone service kiosk in a shopping mall. He earns <code class='latex inline'>\displaystyle \$ 8.50 / \mathrm{h} </code>, plus a <code class='latex inline'>\displaystyle \$ 15 </code> commission for each 1-year service contract he sells.</p><p>a) Find the amount Murray makes in <code class='latex inline'>\displaystyle 8 \mathrm{~h} </code> when he sells seven service contracts.</p><p>b) How many service contracts does Murray need to sell to earn <code class='latex inline'>\displaystyle \$ 790 </code> in a 40 -h work week?</p>

<p>Marcel has <code class='latex inline'>\$40</code> to spend on amusement park rides. Tickets cost <code class='latex inline'>\$1.50</code> without a special membership pass, or <code class='latex inline'>\$1.25</code> with a membership pass. A membership pass costs <code class='latex inline'>\$5.00</code>. Should Marcel buy a membership pass? How many more or less number of rides can Marcel go on with membership pass?</p>

<p>A speed of <code class='latex inline'>\displaystyle 75 \mathrm{mi} / \mathrm{h} </code> is equal to a speed of <code class='latex inline'>\displaystyle 110 \mathrm{ft} / \mathrm{s} </code>. To the nearest mile per hour, what is the speed of an aircraft traveling at a speed of <code class='latex inline'>\displaystyle 1600 \mathrm{ft} / \mathrm{s} ? </code></p>

<p>Heinrich and his brother live in Germany.
They are taking a trip to the United States and have been checking the average temperatures in different U.S. cities for the month they will be traveling. They are unfamiliar with the Fahrenheit scale, so they would like to convert the temperatures to Celsius. The equation <code class='latex inline'>F=1.8C+32</code> relates the temperature in degrees Celsius C to degrees Fahrenheit F.</p><p>Find the temperatures in degrees Celsius for each city.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|}
\hline { City } & Temperature \left({ }^{\circ} F\right) \\
\hline New York & 34 \\
\hline Chicago & 23 \\
\hline San Francisco & 55 \\
\hline Miami & 72 \\
\hline Washington, D.C. & 40 \\
\hline
\end{array}
</code></p>

<p>A group of musicians who made a CD are paid according to the
following breakdown, where <code class='latex inline'>n</code> is the number of CDs sold.</p><img src="/qimages/1553" />
<ul>
<li>Find a simplified expression for the total amount paid
to the group.</li>
</ul>

<p>An ice sculpture in the form of a tower is melting at a constant rate of 4cm/h. The ice sculpture is 40 cm high when it first starts to melt. </p><p>a) Set up a graph of height, <code class='latex inline'>h</code>, in centimetres, versus time, <code class='latex inline'>t</code>, in hours, and plot the<code class='latex inline'>h</code>-intercept. </p><p>b) Should the slope of this linear relation be positive or negative? Explain. </p><p>c) Graph the line. </p><p>d) What is the height of the ice sculpture after </p>
<ul>
<li>4 hours</li>
<li>5.5 hours?</li>
</ul>
<p>e) Identify the <code class='latex inline'>t</code>-intercept and explain what it means. </p><p>f) Explain why this graph has no meaning below the <code class='latex inline'>t</code>-axis. </p>

<p>Mr. Martinez is a sales representative for an agricultural supply company. He receives a salary and monthly commission. He also receives a bonus each time he reaches a sales goal.</p><p>Write a verbal expression that describes how much Mr. Martinez earns in a year if he receives four equal bonuses.</p>

<p>A driver drove 12 miles and made a pit stop. After that, the driver continued driving at a constant speed of 65 miles per hour for <code class='latex inline'>\displaystyle t </code> hours. Which of the following represents the total distance driven?</p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { (F) } 12+65 t & \text { (G) } 65 t & \text { (H) } 12 t+65 & \text { D } 12(t+65)\end{array} </code></p>

<p>Erin joins a CD club. The first 10 CDs are free, but after that she pays <code class='latex inline'>\$15.95</code> for each CD she orders.</p>
<ul>
<li>How much would she pay for 15 CDs?</li>
</ul>

<img src="/qimages/43316" />

<p>The amount of blood in the body can be predicted by the equation <code class='latex inline'>y=0.07w</code>, where <code class='latex inline'>y</code> is the number of pints of blood and <code class='latex inline'>w</code> is the weight of a person in pounds.</p><p>Predict the weight of a person whose body holds 12 pints of blood.</p>

<p>Use the table below that shows the maximum heart rate to maintain, for different ages, during aerobic activities such as running, biking, or swimming.</p><img src="/qimages/24684" /><p>Write an equation in function notation for the relation.</p>

<p>Alison has a part-time job as a lifeguard. Alison‘s pay varies directly with the time, in hours, she works. She earns $9.75/h.</p><p>a) Explain why this relationship is considered a direct variation.</p><p>b) Write an equation representing Alison‘s regular pay.</p><p>c) Graph this relationship, using pencil and paper or technology.</p>

<p>A marina charges $9.50 per hour to rent a boat.</p><p>a) Describe the relationship between the cost of the boat rental and the time, in hours, the boat is rented for.</p><p>b) Illustrate the relationship graphically and represent it with an equation.</p><p>c) Use your graph to estimate the cost of renting the boat for 12 h.</p><p>d) Use your equation to determine the exact cost of renting the boat for
12 h.</p>

<p>Chapter Problem Your first task as producer of Canadian Superstar is to rent a theatre for the first event, a singing competition. Rental includes lunch and snacks for the competitors. Details for the two best choices are shown:</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|}
\hline Royal james Hall \$50 per person \\
\hline
\end{array}
</code> <code class='latex inline'>\displaystyle
\begin{array}{|l|}
\hline Broadway Nights \\
\$ 1000 plus \$ 30 per person \\
\hline
\end{array}
</code> 10</p><p>You have <code class='latex inline'>\displaystyle \$ 2000 </code> in your budget for this event. You would like to begin the competition with as many contestants as you can afford.</p><p>The cost for renting Broadway Nights can be modelled by the equation <code class='latex inline'>\displaystyle C=1000+30 n </code>. Explain why this equation correctly gives the cost, <code class='latex inline'>\displaystyle C </code>, in dollars, for <code class='latex inline'>\displaystyle n </code> contestants.</p>

<p>Think About a Plan Suppose you make a 4-minute local call using a calling card and are charged <code class='latex inline'>\displaystyle 7.6 </code> cents. The cost of a local call varies directly with the length of the call. How much more will it cost to make a 30-minute local call?</p>
<ul>
<li><p>Which quantity is the dependent quantity?</p></li>
<li><p>How does the word "more" affect the method needed to solve the problem?</p></li>
</ul>

<p>Determine whether <code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hlinex & y \\ \hline 1 & -2 \\ \hline 3 & -8 \\ \hline 5 & 14 \\ \hline\end{array} </code></p>

<p>John was bringing a message to the principal's office when the principal intercepted him and took the message. When a graph passes through the <code class='latex inline'>\displaystyle y </code> -axis, it has a y-intercept. What do you think a <code class='latex inline'>\displaystyle y </code> -intercept of a graph represents?</p>

<p>A math textbook costs <code class='latex inline'>\displaystyle \$ 60.00 . </code> The number of students who need the book is represented by <code class='latex inline'>\displaystyle x </code>. The total cost of purchasing books for a group of students can be represented by the function <code class='latex inline'>\displaystyle f(x) </code>.</p><p>a) Write an equation in function notation to represent the cost of purchasing textbooks for <code class='latex inline'>\displaystyle x </code> students.</p><p>b) State the degree of this function and whether it is linear or quadratic.</p><p>c) Use your equation to calculate the cost of purchasing books for a class of 30 students.</p><p>d) What are the domain and range of this function, assuming that books can be purchased for two classes of students? Assume that the maximum number of students in a class is <code class='latex inline'>\displaystyle 30 . </code> Express your answers in set notation.</p>

<p>At the season finale, you present the winner of Canadian Superstar with a recording-and-tour contract. The contract states that the winer will be paid <code class='latex inline'>\$ 5000</code> per month while on tour plus <code class='latex inline'>\$2</code> per CD sold.</p><p>How much will the winner earn after the first month if 500 CDs are sold?</p>

<p>If <code class='latex inline'>\displaystyle x </code> is doubled, what happens to <code class='latex inline'>\displaystyle y </code> ?</p>

<p>Heinrich and his brother live in Germany.
They are taking a trip to the United States and have been checking the average temperatures in different U.S. cities for the month they will be traveling. They are unfamiliar with the Fahrenheit scale, so they would like to convert the temperatures to Celsius. The equation <code class='latex inline'>F=1.8C+32</code> relates the temperature in degrees Celsius C to degrees Fahrenheit F.</p><p>Solve the equation for <code class='latex inline'>C</code>.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|}
\hline { City } & Temperature \left({ }^{\circ} F\right) \\
\hline New York & 34 \\
\hline Chicago & 23 \\
\hline San Francisco & 55 \\
\hline Miami & 72 \\
\hline Washington, D.C. & 40 \\
\hline
\end{array}
</code></p>

<p><code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>.</p><p>If <code class='latex inline'>\displaystyle y=4 </code> when <code class='latex inline'>\displaystyle x=-2 </code>, find <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=6 </code>.</p>

<p>Kwok is a hotel manager. His responsibilities include renting rooms
for conferences. The hotel charges $250 per day plus $15 per person for the grand ballroom.</p>
<ul>
<li>Create a formula that relates the cost, <code class='latex inline'>C</code>, in dollars, of renting the ballroom to the number of people, <code class='latex inline'>n</code>.</li>
</ul>

<p>Open-Ended Choose a value of <code class='latex inline'>\displaystyle k </code> within the given range. Then write and graph a direct variation function using your value for <code class='latex inline'>\displaystyle k </code>.</p><p><code class='latex inline'>\displaystyle -1 < k < -\frac{1}{2} </code></p>

<p>Open-Ended Choose a value of <code class='latex inline'>\displaystyle k </code> within the given range. Then write and graph a direct variation function using your value for <code class='latex inline'>\displaystyle k </code>.</p><p><code class='latex inline'>\displaystyle 0 < k < 1 </code></p>

<p>Paloma works part-time, 4h per day, selling fitness club memberships. She is paid <code class='latex inline'>\$9/h</code>, plus a <code class='latex inline'>\$12</code> commission for each 1-year memberships she sells.</p><p>How many memberships does Paloma need to sell to earn <code class='latex inline'>\$600</code> in a <code class='latex inline'>24</code>-h workweek?</p>

<p>Find the rate of change for each set of data.</p><img src="/qimages/31637" />

<p>You can use the formula <code class='latex inline'>C =2.5I</code> to obtain an approximate value for converting a length, <code class='latex inline'>I</code>, in inches to a length, <code class='latex inline'>C</code>, in centimetres.</p>
<ul>
<li>Rearrange the formula to express <code class='latex inline'>I</code> in terms of <code class='latex inline'>C</code>.</li>
<li><p>How many inches are in </p></li>
<li><p>75 cm?</p></li>
<li><p>1 m?</p></li>
</ul>

<p>The circumference, <code class='latex inline'>C</code>, in kilometres, of the tropical storm in question 5 can be modelled by the function <code class='latex inline'>C(r) =2\pi r</code>.</p><p><strong>a)</strong> Graph <code class='latex inline'>C(r)</code> for <code class='latex inline'>r\in [0, 10]</code></p><p><strong>b)</strong> State the domain and range.</p><p><strong>c)</strong> Describe the similarities and differences between the graph of <code class='latex inline'>C(r)</code> and the graph of <code class='latex inline'>y = x</code>.</p>

<p>A plumber charges according to the equation <code class='latex inline'>60n - C + 90 = 0</code>, where <code class='latex inline'>C</code> is the total charge, in dollars, for a house call, and <code class='latex inline'>n</code> is the time, in hours, the job takes.</p><p>a) Rearrange the equation to express it in the form <code class='latex inline'>C = mn + b</code>.</p><p>b) Identify the slope and the C-intercept and explain what they mean.</p><p>c) Graph the relation.</p><p>d) What would a 3-h house call cost?</p>

<p>Which equation best represents the</p><p>relationship between the number of hours an electrician works <code class='latex inline'>\displaystyle h </code> and the total charges <code class='latex inline'>\displaystyle c </code> ? </p><p> A <code class='latex inline'>\displaystyle c=30+55 </code></p><p>B <code class='latex inline'>\displaystyle c=30 h+55 </code></p><p>C <code class='latex inline'>\displaystyle c=30+55 h </code></p><p>D <code class='latex inline'>\displaystyle c=30 h+55 h </code></p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|l|}
\hline { \text{ Cost of Electrician} } \\
\hline \text{ Emergency House Call} & \$ 30 \text{ one time fee} \\
\hline \text{ Rate} & \$ 55 / \text{ hour} \\
\hline
\end{array}
</code></p>

<p>As a thunderstorm approaches, you see lightning as it occurs, but you hear the accompanying sound of thunder a short time afterward. The distance <code class='latex inline'>d</code> in miles that sound travels in <code class='latex inline'>t</code> seconds is given by the equation <code class='latex inline'>d=0.21t</code>.</p><p>Graph the equation.</p>

<p>As a thunderstorm approaches, you see lightning as it occurs, but you hear the accompanying sound of thunder a short time afterward. The distance <code class='latex inline'>d</code> in miles that sound travels in <code class='latex inline'>t</code> seconds is given by the equation <code class='latex inline'>d=0.21t</code>.</p><p>Estimate how long it will take to hear the thunder from a storm 3 miles away.</p>

<p>Shopping You are buying orange juice for <code class='latex inline'>\displaystyle \$ 4.50 </code> per container and have a gift card worth <code class='latex inline'>\displaystyle \$ 7 . </code> The function <code class='latex inline'>\displaystyle f(x)=4.50 x-7 </code> represents your total cost <code class='latex inline'>\displaystyle f(x) </code> if you buy <code class='latex inline'>\displaystyle x </code> containers of orange juice and use the gift card. How much do you pay to buy 4 containers of orange juice?</p>

<p>Identify the slope and the vertical intercept of each linear relation and explain what they represent. Write an equation to describe the relationship.</p><img src="/qimages/1203" />

<p>A film club sponsors a film fest at a local movie theater. Renting the theater costs
<code class='latex inline'> \$ 190 </code> . The admission is <code class='latex inline'> \$ 2 </code> per person.</p><p>a. Write an equation that relates the film club's total cost <code class='latex inline'> c </code> and the number of
people <code class='latex inline'> p </code> who attend the film fest.</p><p>b. Graph the equation you wrote in part (a).</p>

<p>ROLLER COASTERS The speed of the Steel Dragon 2000 roller coaster in Mie Prefecture, Japan, can be modeled by <code class='latex inline'>\displaystyle y=10.4 x </code>, where <code class='latex inline'>\displaystyle y </code> is the distance traveled in meters in <code class='latex inline'>\displaystyle x </code> seconds.</p><p>a. How far does the coaster travel in 25 seconds?</p><p>b. The speed of the Kingda Ka roller coaster in Jackson, New Jersey, can be described by <code class='latex inline'>\displaystyle y=33.9 x </code>. Which coaster travels faster? Explain your reasoning.</p>

<p>a) Copy and complete the table of values given that y varies partially with x.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline x & y \\
\hline 0 & -3 \\
\hline 1 & 1 \\
\hline 2 & \\
\hline 3 & 9 \\
\hline 4 & \\
\hline & 29 \\
\hline
\end{array}
</code></p><p>b) Identify the initial value of <code class='latex inline'>y</code> and the constant of variation from the table.</p><p>c) Write an equation relating y and x in the form <code class='latex inline'>y = mx + b</code>.</p><p>d) Graph the relation. Describe the graph.</p>

<p>The height of a tree is related to the diameter of its trunk. The table shows the height and the diameter of five maple trees.</p><p><code class='latex inline'>\displaystyle
\begin{array}{l|r|r|r|r|r}
Diameter (\mathrm{cm}) & 112 & 120 & 122 & 132 & 140 \\
\hline Height (m) & 27 & 28 & 29 & 31 & 33
\end{array}
</code></p><p>a. State the ordered pairs <code class='latex inline'>\displaystyle (D, H) </code> of this relation.</p><p>b. State the domain and range of this relation.</p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (1,2) </code></p>

<p>Find the rate of change for each set of data.</p><img src="/qimages/31636" />

<p>Since pure gold is very soft, other metals are often added to it to make an alloy that is stronger and more durable. The relative amount of gold in a piece of jewelry is measured in karats. The formula for the relationship is <code class='latex inline'>g=\frac{25k}{6}</code>, where <code class='latex inline'>k</code> is the number of karats and <code class='latex inline'>g</code> is the percent of gold in the jewelry.</p><p>Find the percent of gold if the domain is {10, 14, 18, 24}. Make a table of values and graph the function.</p>

<p>Alberto is training for a triathlon, where athletes swim, cycle, and run. During his training program, he has found that he can swim at 1.2 km/h, cycle at 25 km/h, and run at 10 km/h. To estimate his time for an upcoming race, Alberto rearranges the formula
<code class='latex inline'>distance = speed \times time</code> to find that: </p><p><code class='latex inline'>\displaystyle
time = \frac{distance}{speed}
</code>.</p><p>a) Choose a variable to represent the distance travelled for each part of the race. For example, choose s for the swim.</p><p>b) Copy and complete the table. The first row is done for you.</p><img src="/qimages/10106" /><p>c) Write a trinomial to model Alberto’s total time.</p><p>d) A triathlon is advertised in Kingston. Participants have to swim 1.5 km, cycle 40 km, and run 10 km. Using your expression from part c), calculate how long it will take Alberto to finish the race.</p><p>e) Is your answer a reasonable estimate of Alberto’s triathlon time? Explain.</p>

<p>You can use the formula <code class='latex inline'>\displaystyle L=3.8 G </code> to obtain an approximate value for converting a</p><p>volume in U.S. gallons, <code class='latex inline'>\displaystyle G </code>, to a volume in litres, <code class='latex inline'>\displaystyle L </code>.</p><p>a) Use the formula to find the number of litres in</p>
<ul>
<li><p><code class='latex inline'>\displaystyle 0.5 </code> gallons</p></li>
<li><p>1 pint <code class='latex inline'>\displaystyle (1 </code> pint <code class='latex inline'>\displaystyle =0.125 </code> gallons <code class='latex inline'>\displaystyle ) </code> b) Rearrange the formula to express <code class='latex inline'>\displaystyle G </code> in terms of <code class='latex inline'>\displaystyle L . </code></p></li>
</ul>
<p>c) How many gallons are in</p>
<ul>
<li><p><code class='latex inline'>\displaystyle 4 \mathrm{~L} </code> ?</p></li>
<li><p><code class='latex inline'>\displaystyle 250 \mathrm{~mL} </code> ?</p></li>
</ul>

<p>The function <code class='latex inline'>s(d)=0.159+0.118d</code> relates the slope, <code class='latex inline'>s</code>, of a beach to the average diameter, <code class='latex inline'>d</code>, in millimetres, of the sand particles on the beach. Which beach has a steeper slope: beach <code class='latex inline'>A</code>, which has very fine sand with <code class='latex inline'>d=0.0625</code>, or beach <code class='latex inline'>B</code>, which has very coarse sand with <code class='latex inline'>d=1?</code> Justify your decision. </p>

<p>Determine the constant of variation for the direct variation.</p><p>The money earned by an employee varies directly with time. The employee earned $320 in 40 h.</p>

<p>REASONING Recall that the standard form of a linear equation is <code class='latex inline'>\displaystyle A x+B y=C . </code> Rewrite this equation in slope-intercept form. Use your answer to find the slope and <code class='latex inline'>\displaystyle y </code> -intercept of the graph of the equation <code class='latex inline'>\displaystyle -6 x+5 y=9 </code></p>

<p>You and some friends are going to a museum. Each ticket costs $4.50.</p><p>a. If <code class='latex inline'> n </code> is the number of tickets purchased, write an expression that gives the total
cost of buying <code class='latex inline'> n </code> tickets.</p><p>b. Suppose the total cost for <code class='latex inline'> n </code> tickets is <code class='latex inline'> \$ 36 </code> . What is the total cost if one more ticket
is purchased?</p>

<p>A swimming pool manufacturer installs rectangular pools whose length is twice the width, plus 5 m.</p><p>Calculate the new amount required and compare this with your prediction. Explain the results.</p>

<p>BABYSITTING Christina earns <code class='latex inline'>\displaystyle \$ 7.50 </code> an hour babysitting.</p><p>a. Write an algebraic expression to represent the money Christina will earn if she works <code class='latex inline'>\displaystyle h </code> hours.</p><p>b. Choose five values for the number of hours Christina can babysit. Create a table with <code class='latex inline'>\displaystyle h </code> and the amount of money she will make during that time.</p><p>c. Use the values in your table to create a graph.</p><p>d. Does it make sense to connect the points in your graph with a line? Why or why not?</p>

<p>a) A car travels at <code class='latex inline'>\displaystyle 45 \mathrm{~km} / \mathrm{h} </code> for <code class='latex inline'>\displaystyle 2.5 \mathrm{~h} </code>. How far does the car travel?</p><p>b) Rearrange the formula <code class='latex inline'>\displaystyle d=s t </code> to solve for <code class='latex inline'>\displaystyle s </code>. Use this formula to find the speed of a truck that travels <code class='latex inline'>\displaystyle 262.5 \mathrm{~km} </code> in <code class='latex inline'>\displaystyle 3.5 \mathrm{~h} </code>.</p><p>c) Rearrange the formula <code class='latex inline'>\displaystyle d=s t </code> to solve for <code class='latex inline'>\displaystyle t </code>. Use this formula to find how long it would take a boat to travel <code class='latex inline'>\displaystyle 59.5 \mathrm{~km} </code> at a speed of <code class='latex inline'>\displaystyle 34 \mathrm{~km} / \mathrm{h} </code></p>

<p>Three authors team up to write a children’s book. The publisher pays them according to the following contracts.</p><img src="/qimages/1140" /><p>(b) Determine the total payout if the book sells</p>
<ul>
<li>200 copies</li>
<li>5000 copies</li>
</ul>

<p>The number of tourists visiting a small seaside town each summer is decreasing. This year the number of visitors was 3400, and it has been predicted that every year there will be 260 fewer tourists.</p><p>a) Write an explicit formula to determine the number of tourists in any given year.</p><p>b) How many tourists are expected to visit in 6 years?</p><p>c) How long will it be before the number of tourists drops below 1500?</p>

<p>A hang glider, 25 meters above the ground, starts to descend at a constant rate of 2 meters per second. Which equation shows the height <code class='latex inline'>\displaystyle h </code> after <code class='latex inline'>\displaystyle t </code> seconds of descent? A <code class='latex inline'>\displaystyle h=25 t+2 t </code></p><p>B <code class='latex inline'>\displaystyle h=-25 t+2 </code></p><p>C <code class='latex inline'>\displaystyle h=2 t+25 </code></p><p>D <code class='latex inline'>\displaystyle h=-2 t+25 </code></p>

<p><code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>.</p><p>If <code class='latex inline'>\displaystyle y=7 </code> when <code class='latex inline'>\displaystyle x=2 </code>, find <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=3 </code>.</p>

<p>Graph each equation.</p><p><code class='latex inline'>\displaystyle 2 x-\frac{3}{2} y=-3 </code></p>

<p>Your friend looks for a pattern in the table below and claims that the output equals the input divided by 2. Is your friend correct? Explain.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|c|c|c|c|}
\hline Input & 3 & 6.8 & 8 & 10 & 25 \\
\hline Output & 2 & 3.4 & 4 & 5 & 12.5 \\
\hline
\end{array}
</code></p>

<p>Without graphing, tell whether the slope of a line that models each linear relationship is positive, negative, zero, or undefined. Then find the slope.</p><p>The length of a bus route is <code class='latex inline'>\displaystyle 4 \mathrm{mi} </code> long on the sixth day and <code class='latex inline'>\displaystyle 4 \mathrm{mi} </code> long on the seventeenth day.</p>

<img src="/qimages/43242" /><p>MULTIPLE REPRESENTATIONS The balance</p><p><code class='latex inline'>\displaystyle y </code> (in dollars) of your savings account is a function of the month <code class='latex inline'>\displaystyle x </code>.</p><p>Month, <code class='latex inline'>\displaystyle x </code> & 0 & 1 & 2 & 3 & 4</p><p>Balance (dollars), <code class='latex inline'>\displaystyle y </code> & 100 & 125 & 150 & 175 & 200</p><p>a. Describe this situation in words.</p><p>b. Write the function as a set of ordered pairs.</p><p>c. Plot the ordered pairs in a coordinate plane.</p>

<p>A taxi company charges a fare of $2.25 plus $0.75 per mile traveled. The cost of the fare <code class='latex inline'>c</code> can be described by the equation <code class='latex inline'>c=0.75m+2.25</code>, where <code class='latex inline'>m</code> is the number of miles traveled.</p><p>Graph the equation.</p>

<p>The piggy bank contained <code class='latex inline'>\displaystyle \$ 25 </code>, and <code class='latex inline'>\displaystyle \$ 1.50 </code> is added each day.</p>

<p>If <code class='latex inline'>\displaystyle y=\frac{1}{2} </code> when <code class='latex inline'>\displaystyle x=4 </code>, find <code class='latex inline'>\displaystyle y </code> when <code class='latex inline'>\displaystyle x=5 </code>.</p>

<p>Describe a situation that could be illustrated by the graph below.</p><img src="/qimages/21735" />

<p>A promoter is holding a video dance. Tickets cost <code class='latex inline'>\$15</code> per person, and he has given away 10 free tickets to radio stations. </p><p>Find how many people bought the ticket if he made <code class='latex inline'>\$600</code>?</p><p>You may use the equation below. </p><p><code class='latex inline'>
\displaystyle
R = 15n - 150
</code></p>

<p>Paloma works part-time, 4h per day, selling fitness club memberships. She is paid <code class='latex inline'>\$9/h</code>, plus a <code class='latex inline'>\$12</code> commission for each 1-year memberships she sells.</p><p>Find the amount Paloma makes in 8 h when she sells seven memberships.</p>

<p>A parking lot charges $14.50 per day for long-term parking at the airport.</p><p>a) Describe the relationship between the cost of the long-term parking and the time, in days, the car is parked for.</p><p>b) Illustrate the relationship graphically and represent it with an equation.</p><p>c) Use your graph to estimate the cost of parking the car for 6 days.</p><p>d) Use your equation to determine the exact cost of parking the car for 6 days.</p>

<img src="/qimages/43315" />

<p>Think About a Plan The table at the right shows the number of bagels a shop gives you per "baker's dozen." Write an algebraic expression that gives the rule for finding the number of bagels in any number <code class='latex inline'>\displaystyle b </code> of baker's dozens.</p>
<ul>
<li><p>What is the pattern of increase in the number of bagels?</p></li>
<li><p>What operation can you perform on <code class='latex inline'>\displaystyle b </code> to find the number of bagels?</p><img src="/qimages/12401" /></li>
</ul>
<p>Bagels Baker's nom 5 \begin{tabular}{c|c}Baker's Dozens & Number of Bagels \1 & 13 \2 & 26 \3 & 39 \<code class='latex inline'>\displaystyle b </code> &</p>

<p>The equation <code class='latex inline'>\displaystyle d=4-\frac{1}{15} t </code> represents your distance from home <code class='latex inline'>\displaystyle d </code> for each minute you walk <code class='latex inline'>\displaystyle t . </code></p><p>a. If you graphed this equation, what would the slope represent? Explain.</p><p>b. Are you walking towards or away from your home? Explain.</p>

<p>When you buy a car. its value depreciates (becomes. less) over time. The graph illustrates the value of a car from the time it was bought.</p><img src="/qimages/22048" /><p>a) How much did the car originally cost?</p><p>b) After what period of time does the car no longer have any value?</p><p>c) What is the slope of this graph and what does it mean?</p>

<p>If <code class='latex inline'>\displaystyle z </code> varies directly with the product of <code class='latex inline'>\displaystyle x </code> and <code class='latex inline'>\displaystyle y(z=k x y) </code>, then <code class='latex inline'>\displaystyle z </code> is said to vary jointly with <code class='latex inline'>\displaystyle x </code> and <code class='latex inline'>\displaystyle y </code>.</p><p>a. Geometry The area of a triangle varies jointly with its base and height. What is the constant of variation?</p><p>b. Suppose <code class='latex inline'>\displaystyle q </code> varies jointly with <code class='latex inline'>\displaystyle v </code> and <code class='latex inline'>\displaystyle s </code>, and <code class='latex inline'>\displaystyle q=24 </code> when <code class='latex inline'>\displaystyle v=2 </code> and <code class='latex inline'>\displaystyle s=3 . </code> Find <code class='latex inline'>\displaystyle q </code> when <code class='latex inline'>\displaystyle v=4 </code> and <code class='latex inline'>\displaystyle s=2 . </code></p><p>c. Reasoning Suppose <code class='latex inline'>\displaystyle z </code> varies jointly with <code class='latex inline'>\displaystyle x </code> and <code class='latex inline'>\displaystyle y </code>, and <code class='latex inline'>\displaystyle x </code> varies directly with <code class='latex inline'>\displaystyle w . </code> Show that <code class='latex inline'>\displaystyle z </code> varies jointly with <code class='latex inline'>\displaystyle w </code> and <code class='latex inline'>\displaystyle y . </code></p>

<p>Conservation A dripping faucet wastes a cup of water if it drips for three minutes. The amount of water wasted varies directly with the amount of time the faucet drips. How long will it take for the faucet to waste <code class='latex inline'>\displaystyle 4 \frac{1}{2} </code> cups of water?</p>

<p>SNAKES Suppose the body length <code class='latex inline'>\displaystyle L </code> in inches of a baby snake is given by <code class='latex inline'>\displaystyle L(m)=1.5+2 m </code>, where <code class='latex inline'>\displaystyle m </code> is the age of the snake in months until it becomes 12 months old.</p><p>a. Find the length of an 8 -month-old snake.</p><p>b. Find the snake's age if the length of the snake is <code class='latex inline'>\displaystyle 25.5 </code> inches.</p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (-0.1,50) </code></p>

<p>A swimming pool manufacturer installs rectangular pools whose length is twice the width, plus 5 m.</p><p>Predict how the amount of coping will change if you double the width of the pool.</p>

<p>For each function, determine whether y varies directly with x. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{c|c}\hlinex & y \\ \hline 3 & 9 \\ \hline 4 & 10 \\ \hline 5 & 11 \\ \hline\end{array} </code></p>

<img src="/qimages/43245" /><p>OPEN-ENDED Fill in the table so that when <code class='latex inline'>\displaystyle t </code> is the independent variable, the relation is a function, and when <code class='latex inline'>\displaystyle t </code> is the dependent variable, the relation is not a function.</p><p><code class='latex inline'>\displaystyle \boldsymbol{t} </code> & &</p><p><code class='latex inline'>\displaystyle \boldsymbol{v} </code> & & & </p>

<p>An electrician charges according to the equation <code class='latex inline'>\displaystyle 75 n-C+60=0 </code>, where <code class='latex inline'>\displaystyle C </code> is the total charge, in dollars, for a house call, and <code class='latex inline'>\displaystyle n </code> is the time, in hours, the job takes. a) Rearrange this equation to express it in the form <code class='latex inline'>\displaystyle C=m n+b </code>.</p><p>b) Identify the slope and the <code class='latex inline'>\displaystyle C </code>-intercept and explain what they mean.</p><p>c) Graph the relation.</p><p>d) What would a 2-h house call cost?</p>

<p>Describe and correct the error in the statement about the relation shown in the table. </p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|l|l|l|l|l|}
\hline Input, \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\
\hline Output, \boldsymbol{y} & 6 & 7 & 8 & 6 & 9 \\
\hline
\end{array}
</code></p><p>The relation is not a function. One</p><p>output is paired with two inputs.</p>

<p><code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>.</p><p>If <code class='latex inline'>\displaystyle y=-7 </code> when <code class='latex inline'>\displaystyle x=-3 </code>, find <code class='latex inline'>\displaystyle y </code> when <code class='latex inline'>\displaystyle x=9 </code>.</p>

<p> The distance-time graph illustrates a person’s movements in front of
a motion sensor.</p><img src="/qimages/6664" /><p>a) Identify the slope and the d-intercept. Explain what they mean.</p><p>b) Write an equation in the form <code class='latex inline'>\displaystyle
d = mt +b
</code> that describes the walker’s motion.</p>

<p> The graph shows how Sally’s weekly earnings vary with the dollar value of the sales she makes at a clothing store.</p><img src="/qimages/9605" /><p>a) What do the coordinates (1500, 360) mean? </p><p>b) Use the graph to determine what Sally earns when her sales are $3200.</p><p>c) Use the graph to determine what sales Sally
needs to make if she wants to earn $450. </p><p>d) Check your answers for parts b) and C) algebraically.</p>

<p>Create a word problem that can be represented by the equation given.</p>
<ul>
<li> <code class='latex inline'>y = 900 + 0.025x</code></li>
</ul>

<img src="/qimages/1554" />
<ul>
<li>Which musician makes the most money at each level
in the table in part b)?</li>
</ul>

<p>Determine whether <code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{|r|r|}\hlinex & y \\ \hline 9 & 6 \\ \hline 12 & 8 \\ \hline 15 & 10 \\ \hline\end{array} </code></p>

<p>Under water, pressure increases 4.3 pounds per square inch (psi) for every 10 feet you descend. This can be expressed by the equation <code class='latex inline'>p=0.43d+14.7</code>, where <code class='latex inline'>p</code> is the pressure in pounds per square inch and <code class='latex inline'>d</code> is the depth in feet.</p><p>Graph the equation.</p>

<p>a) Copy and complete the table of</p><p>values given that <code class='latex inline'>\displaystyle y </code> varies partially with <code class='latex inline'>\displaystyle x . </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hlinex & y \\ \hline 0 & 5 \\ \hline 1 & 9 \\ \hline 2 & \\ \hline 3 & 17 \\ \hline 4 & \\ \hline & 37 \\ \hline\end{array} </code></p><p>b) Identify the initial value of <code class='latex inline'>\displaystyle y </code> and the constant of variation from the table.</p><p>c) Write an equation relating <code class='latex inline'>\displaystyle y </code> and <code class='latex inline'>\displaystyle x </code> in the form <code class='latex inline'>\displaystyle y=m x+b </code>.</p><p>d) Graph the relation. Describe the</p><p>graph.</p>

<p>FINANCIAL LITERACY Samuel has <code class='latex inline'>\displaystyle \$ 1900 </code> in the bank. He wishes to increase his account to a total of <code class='latex inline'>\displaystyle \$ 2500 </code> by depositing \$30 per week from his paycheck. Write and solve an equation to find how many weeks he needs to reach his goal.</p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (9,-1) </code></p>

<p>Write and graph a direct variation equation that passes through each point.</p><p><code class='latex inline'>\displaystyle (-5,-3) </code></p>

<p>The points represented by the table lie on a line. Find the slope of the line.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline \boldsymbol{x} & \boldsymbol{y} \\
\hline 6 & 9 \\
11 & 15 \\
16 & 21 \\
21 & 27 \\
\hline
\end{array}
</code></p>

<p>Class Project The freshman class will be selling carnations as a class project. What is the class's income after it pays the florist a flat fee of <code class='latex inline'>\displaystyle \$ 200 </code> and sells <code class='latex inline'>\displaystyle x </code> carnations for <code class='latex inline'>\displaystyle \$ 2 </code> each?</p>

<p><code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>.</p><p>If <code class='latex inline'>\displaystyle y=5 </code> when <code class='latex inline'>\displaystyle x=-3 </code>, find <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=-1 </code>.</p>

<p><code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>.</p><p>If <code class='latex inline'>\displaystyle y=6 </code> when <code class='latex inline'>\displaystyle x=2 </code>, find <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=12 </code>.</p>

<img src="/qimages/43365" /><p>PROBLEM SOLVING The graph shows the percent <code class='latex inline'>\displaystyle p </code> (in decimal form) of battery power remaining in a laptop computer after <code class='latex inline'>\displaystyle t </code> hours of use. A tablet computer initially has <code class='latex inline'>\displaystyle 75 \% </code> of its battery power remaining and loses <code class='latex inline'>\displaystyle 12.5 \% </code> per hour. Which computer's battery will last longer? Explain. (See Example 5.)</p>

<p>If <code class='latex inline'>\displaystyle y=\frac{5}{3} </code> when <code class='latex inline'>\displaystyle x=\frac{3}{4} </code>, find <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=\frac{1}{2} </code></p>

<p>Protect-a-Boat Insurance Company charges $400 for liability, plus 15% of the value of the boat, plus $200 per passenger.</p><p>a) Write an expression to model the insurance cost.</p><p>b) Find the cost of insurance tor a $120 000 boat that can carry 60 passengers.</p>

<p>In 1898 , A.E. Dolbear studied various species of crickets to determine their "chirp rate" based on temperatures. He determined that the formula <code class='latex inline'>\displaystyle t=50+\frac{n-40}{4} </code>, where <code class='latex inline'>\displaystyle n </code> is the number of chirps per minute, could be used to find the temperature <code class='latex inline'>\displaystyle t </code> in degrees Fahrenheit. What is the temperature if the number of chirps is 120 ?</p>

<p>At the season finale, you present the winner of Canadian Superstar with a recording-and-tour contract. The contract states that the winer will be paid <code class='latex inline'>\$ 5000</code> per month while on tour plus <code class='latex inline'>\$2</code> per CD sold.</p><p>Suppose after the third month on your the new recording artist has earned a total of \$74 000. How many CDs were sold?</p>

<p>Determine whether <code class='latex inline'>\displaystyle y </code> varies directly with <code class='latex inline'>\displaystyle x </code>. If so, find the constant of variation and write the function rule.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hlinex & y \\ \hline 4 & 1 \\ \hline 6 & 2 \\ \hline 8 & 3 \\ \hline\end{array} </code></p>

<p>If <code class='latex inline'>\displaystyle y=-\frac{5}{8} </code> when <code class='latex inline'>\displaystyle x=\frac{3}{2} </code>, find <code class='latex inline'>\displaystyle x </code> when <code class='latex inline'>\displaystyle y=\frac{2}{5} </code>.</p>

<p>Describe a situation that might lead to this graph.</p><img src="/qimages/21741" />

<p>The cost of a certain type of cookies varies directly with the number of packages of cookies that are purchased. The cookies cost $3.50/package.</p><p>a) Choose appropriate letters for Variables. Make a table of values showing the cost of </p><p>0 packages, 1 package. 2 packages, 3 packages, and 4 packages.</p><p>b) Graph the relationship.</p><p>c) Write an equation for the relationship in the form <code class='latex inline'>y = kx</code>.</p>

<p>There are a lot of factories in the Jean's home city. The equation <code class='latex inline'>n-E+15=0</code> describes how much a worker might earn, <code class='latex inline'>E</code>, in dollars per hours, according to the number of years experience, <code class='latex inline'>n</code>.</p><p>Find the hourly earnings of a beginning factory worker, and of a worker with 5 years of experience, and you will have two more letters in the name of this city.</p>

<p>Jocelyn makes <code class='latex inline'>\displaystyle x </code> dollars per hour working at the grocery store and <code class='latex inline'>\displaystyle n </code> dollars per hour babysitting. Write an expression that describes her earnings if she babysat for 25 hours and worked at the grocery store for 15 hours.</p>

<p>A health club offers two types of monthly memberships:</p>
<ul>
<li>Membership A: $3 per visit</li>
<li>Membership B: a flat fee of $8 and $2 per visit</li>
</ul>
<p>a) Graph both relations for 0 to 10 visits.</p><p>b) Classify each relation as a direct variation or a partial variation.</p><p>c) Write an equation relating the cost and the number of visits for each membership.</p><p>d) Compare the monthly membership costs. When is Membership A cheaper than Membership B? When is Membership B cheaper than
Membership A?</p>

<p>Make a table of <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-values and use it to graph the direct variation equation.</p><p><code class='latex inline'>\displaystyle x=\left(-\frac{1}{3}\right) y </code></p>

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