5. Q5b
Save videos to My Cheatsheet for later, for easy studying.
Video Solution
Q1
Q2
Q3
L1
L2
L3
Similar Question 1
<p>Find all the critical numbers of the function.</p><p><code class='latex inline'>\displaystyle y = \frac{x^2 -1}{x^2 +1} </code></p>
Similar Question 2
<p>Describe the end behavior of the graph shown.</p><img src="/qimages/38424" />
Similar Question 3
<p>Determine the equations of any vertical or horizontal asymptotes for each function. Describe the behaviour of the function on each side of any vertical or horizontal asymptote.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{x-5}{2x + 1} </code></p>
Similar Questions
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>For each function below,</p><p>i) determine <code class='latex inline'>\displaystyle f(-2), f(1) </code>, and <code class='latex inline'>\displaystyle f\left(\frac{1}{2}\right) </code> ii) write the function in mapping</p><p>notation</p><p>e) <code class='latex inline'>\displaystyle f(x)=\frac{1}{4 x+1} </code></p>
<p>For each function <code class='latex inline'>f(x)</code>, determine the equation for <code class='latex inline'>g(x)</code>.</p><p><code class='latex inline'>f(x) = \frac{1}{x - 8} + 3</code><br> <code class='latex inline'>g(x) = -f(x)</code></p>
<p>Sketch each set of functions on the same set of axes.</p><p><code class='latex inline'>y = \frac{1}{x}, y = \frac{2}{x}, y = - \frac{2}{x}, y = -\frac{2}{x-1} + 3</code></p>
<p>For each function g(x), describe the transformation from a base function of <code class='latex inline'>f(x) = x</code>, <code class='latex inline'>f(x) = x^2</code>, <code class='latex inline'>f(x) = \sqrt x</code>, or <code class='latex inline'>f(x) = \dfrac{1}{x}</code>. Then transform the graph of f(x) to sketch a graph of g(x).</p><p><code class='latex inline'>g(x) = \dfrac{6}{x}</code></p>
<p>Which statements are true about the graph of the function</p><p><code class='latex inline'>\displaystyle y = \frac{(x +1)^2}{2x^2 + 5x + 3} </code>?</p><p>i) The x-intercept is <code class='latex inline'>-1</code></p><p>ii) There is a vertical asymptote at <code class='latex inline'>x = -1</code></p><p>iii) There is a horizontal asymptote at <code class='latex inline'>y = \frac{1}{2}</code></p><p>A. i) only</p><p>B. iii) only</p><p>C. i) and iii) only</p><p>D. ii) and iii) only</p><p>E. i, ii), and iii)</p>
<p>Find the missing values for <code class='latex inline'>y</code>.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{1}{x + 5}</code></p> <ul> <li>as <code class='latex inline'>x \to -5^- </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to -5^+ </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to \infty </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to -\infty </code>, <code class='latex inline'>y \to</code></li> </ul>
<p>The time required to fly from one location to another in inversely proportional to the average speed. When the average speed to fly from Quebec City to Vancouver is 350 km/h, the flying time is 11h.</p> <ul> <li>Sketch a graph of this function.</li> </ul>
<p>Sketch the graph of the function <code class='latex inline'>k(x)=-f(-x)</code>. Then, state the domain and range of each function.</p><img src="/qimages/805" />
<p>Describe the reflection that transforms <code class='latex inline'>f(x)</code> into <code class='latex inline'>g(x)</code>.</p><img src="/qimages/24166" />
<p>Copy each graph of <code class='latex inline'>f(x)</code> and sketch its reflection in the <code class='latex inline'>x</code>-axis, <code class='latex inline'>g(x)</code>. Then, state the domain and range of each function. </p><img src="/qimages/805" />
<p>State the domain and the range of each relation.</p><img src="/qimages/21836" />
<p>What is the excluded value for <code class='latex inline'>\displaystyle y=\frac{4}{x+1} ? </code></p>
<p>Find the missing values for <code class='latex inline'>y</code>.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{1}{x - 2}</code></p> <ul> <li>as <code class='latex inline'>x \to 2^- </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to 2^+ </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to \infty </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to -\infty </code>, <code class='latex inline'>y \to</code></li> </ul>
<p>Determine a possible equation to repent each function shown.</p><img src="/qimages/965" />
<p>For each function, state equations for any vertical asymptotes.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{2x -3}{2x -4} </code></p>
<p>Describe the end behavior of the graph shown.</p><img src="/qimages/38424" />
<p>Determine the equation in the form <code class='latex inline'>f(x) = \frac{1}{kx - c}</code> for the function with a vertical asymptote at <code class='latex inline'>x = -1</code> and a <code class='latex inline'>y</code>-intercept at <code class='latex inline'>-0.25</code>.</p>
<p>Determine the equations of nay horizontal asymptotes. Then state whether the curve approaches the asymptote from above or below.</p><p><code class='latex inline'>\displaystyle g(x) = \frac{x^2 + 2x -15}{9-x^2} </code></p>
<p>Determine the equations of nay horizontal asymptotes. Then state whether the curve approaches the asymptote from above or below.</p><p><code class='latex inline'>\displaystyle g(x) = \frac{2x^2 + x + 1}{x + 4} </code></p>
<p>State any restrictions for the function.</p><img src="/qimages/16998" />
<p>Sketch the function.</p><p><code class='latex inline'> \displaystyle f(x) = \frac{1}{\sqrt{x}} </code></p>
<p>What is the domain of the function</p><p><code class='latex inline'>\displaystyle f(x)=\frac{2}{5-x} ? </code></p><p>a) <code class='latex inline'>\displaystyle \{x \in \mathbf{R} \mid x \neq-5\} \quad </code> c) <code class='latex inline'>\displaystyle \{x \in \mathbf{R} \mid x \neq 0\} </code> b) <code class='latex inline'>\displaystyle \{x \in \mathbf{R} \mid x \neq 5\} \quad </code> d) <code class='latex inline'>\displaystyle \{x \in \mathbf{R}\} </code></p>
<p><strong>(a)</strong> Investigate a variety of functions the form <code class='latex inline'>\displaystyle f(x) = \frac{1}{bx + 2}</code>.</p><p><strong>(b)</strong> What is the effect on the graph when the value of <code class='latex inline'>b</code> is varied?</p>
<p>The time required to fly from one location to another in inversely proportional to the average speed. When the average speed to fly from Quebec City to Vancouver is <code class='latex inline'>350 km/h</code>, the flying time is <code class='latex inline'>11h</code>.</p><p><strong>i.</strong> How long would the trip from Quebec to Vancouver take at an average speed of <code class='latex inline'>500 km/h</code>?</p><p><strong>ii.</strong> Describe the rate of change of the time as the average speed increases.</p>
<p>State the domain and range of each function.</p><p><code class='latex inline'>f(x) = \frac{1}{x}</code></p>
<p>Determine algebraically whether <code class='latex inline'>g(x)</code> is a reflection of <code class='latex inline'>f(x)</code>. Check by graphing. </p><p><code class='latex inline'>f(x) = \frac{1}{x} - 9</code><br> <code class='latex inline'>g(x) = -\frac{1}{x} - 9</code></p>
<p>Determine the domain and range.</p><p><code class='latex inline'> \displaystyle h(x)=\frac{1}{x^2} </code></p>
<p>Determine algebraically whether <code class='latex inline'>g(x)</code> is a reflection of <code class='latex inline'>f(x)</code> in each case. Verify your answer by graphing. </p><p><code class='latex inline'>f(x)=\frac{1}{x+7}</code>, <code class='latex inline'>g(x)=\frac{1}{-x+7}</code></p>
<p>For each function <code class='latex inline'>f(x)</code>, determine the equation for <code class='latex inline'>g(x)</code>.</p><p><code class='latex inline'>f(x) = \frac{1}{x - 6} + 5</code><br> <code class='latex inline'>g(x) = -f(-x)</code></p>
<p>The time required to fly from one location to another in inversely proportional to the average speed. When the average speed to fly from Quebec City to Vancouver is 350 km/h, the flying time is 11h.</p> <ul> <li>Write a function to represent the time as a function of speed.</li> </ul>
<p>For each function <code class='latex inline'>f(x)</code>, determine the equation for <code class='latex inline'>g(x)</code>. </p><p><code class='latex inline'>f(x)=\frac{1}{x-3}-6</code>, <code class='latex inline'>g(x)=-f(-x)</code></p>
<p>Determine the domain and the range of each relation.</p><p><code class='latex inline'>\displaystyle y = \frac{2}{5 -x} </code></p>
<p>Bob uses the relationship <code class='latex inline'>Time = \frac{Distance}{Speed}</code> to plan his kayaking trips. Tomorrow Bob plans to kayak 20 km across a calm lake. He wants to graph the relation <code class='latex inline'>T(s) = \frac{20}{s}</code> to see how the time, <code class='latex inline'>T</code>, it will take varies with his kayaking speed, <code class='latex inline'>s</code>. The next day, he will kayak 15 km up a river that flows at 3 km/h. He will need the graph of <code class='latex inline'>\displaystyle T(s) = \frac{15}{s - 3}</code> to plan this trip. Use transformations to sketch both graphs.</p>
<p>For each function, identify the base function as one of <code class='latex inline'>f(x) = x, f(x) = x^2, f(x) = \sqrt{x}</code>, or <code class='latex inline'>f(x) = \frac{1}{x}</code>. Sketch the graphs of the base function and the transformed function, and state the domain and range of the functions. </p><p><code class='latex inline'>m(x) = \frac{2}{x-9}</code></p>
<p>For each function, determine the range for the domain {—2, —1, 0, 1. 2}.</p><p><code class='latex inline'>\displaystyle y = \frac{8}{x +5} </code></p>
<p>For each function, state equations for any vertical asymptotes.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{x^2 -4}{x} </code></p>
<p>For the line <code class='latex inline'>y = 2x - 5</code> and find the x-intercept. Analyze the reciprocals of the y-coordinates on either side of the x-intercepts. How do these number relate to the key features of the function <code class='latex inline'>f(x) = \frac{1}{2x- 5}</code>?</p>
<p>The force required to lift an object is inversely proportional to the distance of the force from the fulcrum of a lever. A force of 200 N is required at a point 3 m from the fulcrum to lift a certain object.</p> <ul> <li> What is the effect on the force needed as the distance from the fulcrum is doubled?</li> </ul>
<p>Without using graphing technology, match each equation with its corresponding graph. Explain your reasoning.</p><p><code class='latex inline'> \displaystyle \begin{array}{cccccc} & a) & y= \frac{-1}{x-3} & b) & y=\displaystyle{\frac{x^2-9}{x-3}} & c) &y=\displaystyle{\frac{1}{(x+3)^2}} \\ & d) & y={\frac{x}{(x-1)(x+3)}} & e) & y=\displaystyle{\frac{1}{x^2+5}} & f) &y=\displaystyle{\frac{x^2}{x-3}} \\ \end{array} </code></p><p>A) <img src="/qimages/475" /></p><p>B) <img src="/qimages/476" /></p><p>C) <img src="/qimages/477" /></p><p>D) <img src="/qimages/478" /></p><p>E) <img src="/qimages/479" /></p><p>F) <img src="/qimages/480" /></p>
<p>For each function, state equations for any vertical asymptotes.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{x -1}{x^2 + 2x +1} </code></p>
<p>Find all the critical numbers of the function.</p><p><code class='latex inline'>\displaystyle y = \frac{x^2 -1}{x^2 +1} </code></p>
<p>Complete each table to describe the behaviour of the function as x approaches each key value.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{1}{x - 8}</code></p> <ul> <li>as <code class='latex inline'>x \to 8^- </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to 8^+ </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to \infty </code>, <code class='latex inline'>y \to</code></li> <li>as <code class='latex inline'>x \to -\infty </code>, <code class='latex inline'>y \to</code></li> </ul>
<p>Determine the equation in the form <code class='latex inline'>f(x) = \frac{1}{kx -c}</code> for the function with a vertical asymptote at <code class='latex inline'>x = 1</code> and a y-intercept at <code class='latex inline'> -1</code>.</p>
<p>For each function <code class='latex inline'>g(x)</code>, describe the transformation from a base function of <code class='latex inline'>f(x) = x, f(x) = x^2, f(x) = \sqrt{x}</code>, or <code class='latex inline'>f(x) = \frac{1}{x}</code>. Then skctch a graph of <code class='latex inline'>f(x)</code> and <code class='latex inline'>g(x)</code> on the same axes.</p><p><code class='latex inline'>g(x) = \frac{7}{x}</code></p>
<p>Determine the equations of any vertical or horizontal asymptotes for each function. Describe the behaviour of the function on each side of any vertical or horizontal asymptote.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{x-5}{2x + 1} </code></p>
<p>Write equations to represent the horizontal and vertical asymptotes of each rational function.</p><img src="/qimages/1640" />
<p>Determine a possible equation to repent each function shown.</p><img src="/qimages/965" />
<p>Sketch the function.</p><p><code class='latex inline'>\displaystyle g(x) = \frac{1}{|x|}</code></p>
<p>State any restrictions for each function.</p><img src="/qimages/5223" />
<p>Find the equation of the graph below.</p><img src="/qimages/1640" />
<p>Describe the transformation on the graph of <code class='latex inline'>y=\frac{1}{x}</code> needed to obtain the graph of each of the following: <code class='latex inline'>\displaystyle y=\frac{x+3}{x+1} </code></p>
<p>Determine a possible equation to repent each function shown.</p><img src="/qimages/6686" />
<p>For each function, state equations for any vertical asymptotes.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{x^2 + 1}{x^2 -3x - 10} </code></p>
<p>Determine the equations of nay horizontal asymptotes. Then state whether the curve approaches the asymptote from above or below.</p><p><code class='latex inline'>\displaystyle g(x) = \frac{x^2 -4x - 5}{(x + 2)^2} </code></p>
<p>The force required to lift an object is inversely proportional to the distance of the force from the fulcrum of a lever. A force of 200 N is required at a point 3 m from the fulcrum to lift a certain object.</p> <ul> <li>a) Determine a function to represent the force as a function of the distance.</li> <li>b) Sketch the graph of this function.</li> </ul>
How did you do?
Found an error or missing video? We'll update it within the hour! 👉
Save videos to My Cheatsheet for later, for easy studying.