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Similar Question 1
<p>Think About a Plan Hydra are small freshwater animals. They can double in number every two days in a laboratory tank. Suppose one tank has an initial population of 60 hydra. When will there be more than 5000 hydra?</p> <ul> <li><p>How can a table help you identify a pattern?</p></li> <li><p>What function models the situation?</p></li> </ul>
Similar Question 2
<p>For each pair of functions, which of the characteristics below do the two functions have in common and distinguishes between them?</p><p><code class='latex inline'>f(x) = 2^x \text{ and } g(x) = |x|</code></p>
Similar Question 3
<p>For each exponential function, state the base function, <code class='latex inline'>y = 5^x</code>. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.</p><p><code class='latex inline'>\displaystyle y = (\frac{1}{2})^{\frac{x}{2}} -3 </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>Find the missing rows:</p><p>Functions: <code class='latex inline'>Q(n) = 600(\frac{5}{8})^n</code></p><p>Exponential Growth or Decay?(Pick one): </p><p>Initial Value: </p><p>Growth or Decay Rate: %</p>
<p>For the pair of functions, give a characteristic that the two functions have in common and a characteristic that distinguishes them.</p><p><code class='latex inline'>\displaystyle f(x) =2^x </code> and <code class='latex inline'>\displaystyle g(x) = x </code></p>
<p>For each exponential function, state the base function, <code class='latex inline'>y = 5^x</code>. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.</p><p><code class='latex inline'>\displaystyle y = \frac{-1}{10}(5)^{3x - 9} + 10 </code></p>
<p>Geometry The formula for volume <code class='latex inline'>\displaystyle V </code> of a sphere with radius <code class='latex inline'>\displaystyle r </code> is <code class='latex inline'>\displaystyle V=\frac{4}{3} \pi r^{3} </code>. Find the radius of a sphere as a function of its volume. Rationalize the denominator.</p>
<p>The graph of <code class='latex inline'>\displaystyle y=2^{x} </code> is stretched horizontally by a factor of 5 , and then translated 3 units down. Which of the following is the resulting</p><p>equation?</p><p><code class='latex inline'>\displaystyle \begin{array}{ll}\text { a) } f(x)=2^{5(x-3)} & \text { c) } f(x)=5^{x}-3 \\ \text { b) } f(x)=2^{\frac{1}{(x+3)}} & \text { d) } f(x)=2^{\left(\frac{1}{x} x\right)}-3\end{array} </code></p>
<img src="/qimages/56569" /><p>Compare the rule and the function table below. Which function has the greater value when <code class='latex inline'>\displaystyle x=12 ? </code> Explain.</p><p>Function 1 Function 2</p><p><code class='latex inline'>\displaystyle y=4^{x} </code><code class='latex inline'>\displaystyle x </code> &amp; 1 &amp; 2 &amp; 3 &amp; 4 \<code class='latex inline'>\displaystyle y </code> &amp; 5 &amp; 25 &amp; 125 &amp; 625</p>
<p>A hot cup of coffee cools according to the equation</p><p><code class='latex inline'>\displaystyle T(t) = 69(\frac{1}{2})^{\frac{t}{30}} + 21 </code></p><p>where <code class='latex inline'>T</code> is the temperature in degrees Celsius and <code class='latex inline'>t</code> is the time in minutes.</p><p>a) Which part of the equation indicates that this is an example of exponential decay?</p><p>b) What was the initial temperature of the coffee?</p><p>c) Use your knowledge of transformations to sketch the graph of this function.</p><p>d) Determine the temperature of the coffee, to the nearest degree, after 48 min.</p><p>e) Explain how the equation would change if the coffee cooled faster.</p><p>f) Explain how the graph would change if the coffee cooled faster.</p>
<p>Admissions A new museum had 7500 visitors this year. The museum curators expect the number of visitors to grow by <code class='latex inline'>\displaystyle 5 \% </code> each year. The function <code class='latex inline'>\displaystyle y=7500 \cdot 1.05^{x} </code> models the predicted number of visitors each year after <code class='latex inline'>\displaystyle x </code> years. Graph the function.</p>
<p>Reasoning Is <code class='latex inline'>\displaystyle y=(-2)^{x} </code> an exponential function? Justify vour answer.</p>
<p>Both <code class='latex inline'>\displaystyle f(x)=x^{2} </code> and <code class='latex inline'>\displaystyle g(x)=2^{x} </code> have a domain of all real numbers. List as many characteristics as you can to distinguish the two functions.</p>
<p>Each graph represents the rate of change of a function. Determine a possible equation for the function. </p><img src="/qimages/11348" />
<p>Find the missing rows:</p><p>Functions: <code class='latex inline'>P(n) = 32(0.95)^n</code></p><p>Exponential Growth or Decay?(Pick one): </p><p>Initial Value: </p><p>Growth or Decay Rate: %</p>
<p>Consider the function <code class='latex inline'>y = (sin x)^2</code>.</p><p>a) Predict the y-intercept of the function. Justify your prediction.</p><p>b) Predict the x-intercepts from <code class='latex inline'>0^{\circ}</code> to <code class='latex inline'>720^{\circ}</code>. Justify your predictions.</p><p>c) Predict the maximum value and the minimum value of the function. Justify your predictions.</p><p>d) Predict the range and the amplitude of the function. Justify your predictions.</p><p>e) Use a graph or a graphing calculator to verify your answers to parts a) to d).</p><p>f) Describe the similarities and differences between the graph of <code class='latex inline'>y = (sin x)^2</code> and the graph of <code class='latex inline'>y = sin x</code>.</p>
<p>For <code class='latex inline'>f(n) = (2^3)^n</code> and <code class='latex inline'>g(n) = 8^n</code>, is <code class='latex inline'>f(n) = g(n)</code>? Show your work.</p>
<p>Find the missing rows:</p><p>Functions: <code class='latex inline'>A(x) = 5(3)^x</code></p><p>Exponential Growth or Decay?(Pick one): </p><p>Initial Value: </p><p>Growth or Decay Rate: %</p>
<p>a. Graph the functions <code class='latex inline'>\displaystyle y=x^{2} </code> and <code class='latex inline'>\displaystyle y=2^{x} </code> on the same axes.</p><p>b. What do you notice about the graphs for the values of <code class='latex inline'>\displaystyle x </code> between 1 and <code class='latex inline'>\displaystyle 3 ? </code></p><p>c. Reasoning How do you think the graph of <code class='latex inline'>\displaystyle y=8^{x} </code> would compare to the graphs of <code class='latex inline'>\displaystyle y=x^{2} </code> and <code class='latex inline'>\displaystyle y=2^{x} ? </code></p>
<p>Think About a Plan Hydra are small freshwater animals. They can double in number every two days in a laboratory tank. Suppose one tank has an initial population of 60 hydra. When will there be more than 5000 hydra?</p> <ul> <li><p>How can a table help you identify a pattern?</p></li> <li><p>What function models the situation?</p></li> </ul>
<p>Writing Find the range of the function <code class='latex inline'>\displaystyle f(x)=500 \cdot 1^{x} </code> using the domain <code class='latex inline'>\displaystyle \{1,2,3,4,5\} . </code> Explain why the definition of exponential function states that <code class='latex inline'>\displaystyle b \neq 1 . </code></p>
<p>For each of the following equations, state the parent function and the transformations that were applied. Graph the transformed function.</p><p><code class='latex inline'>\displaystyle f(x) = 2^{-2x} -3 </code></p>
<p>Find the missing rows:</p><p>Functions: <code class='latex inline'>V(t) = 100(1.08)^t</code></p><p>Exponential Growth or Decay?(Pick one): </p><p>Initial Value: </p><p>Growth or Decay Rate: %</p>
<p>An ant colony has 5000 ants on July 1 and doubles every year. This can be expressed as <code class='latex inline'>N = 5000 \times 2^t</code>, where <code class='latex inline'>N</code> represents the number of ants and <code class='latex inline'>t</code> represents time, in years.</p><p>a) Find the number of ants after 2, 3, 4, and 5 years.</p><p>b) What does <code class='latex inline'>t = 0</code> represent in this situation? What does <code class='latex inline'>t = -2</code> represent?</p><p>c) When were there 625 ants? Explain.</p>
<p>Suppose <code class='latex inline'>\displaystyle (0,4) </code> and <code class='latex inline'>\displaystyle (2,36) </code> are on the graph of an exponential function.</p><p>a. Use <code class='latex inline'>\displaystyle (0,4) </code> in the general form of an exponential function, <code class='latex inline'>\displaystyle y=a \cdot b^{x} </code>, to find the value of the constant <code class='latex inline'>\displaystyle a </code>.</p><p>b. Use your answer from part (a) and <code class='latex inline'>\displaystyle (2,36) </code> to find the value of the constant <code class='latex inline'>\displaystyle b </code>.</p><p>c. Write a rule for the function.</p><p>d. Evaluate the function for <code class='latex inline'>\displaystyle x=-2 </code> and <code class='latex inline'>\displaystyle x=4 </code>.</p>
<p>Each graph represents the rate of change of a function. Determine a possible equation for the function. </p><img src="/qimages/11345" />
<p>For each pair of functions, which of the characteristics below do the two functions have in common and distinguishes between them?</p><p><code class='latex inline'>f(x) = 2^x \text{ and } g(x) = |x|</code></p>
<p>Environment A solid waste disposal plan proposes to reduce the amount of garbage each person throws out by <code class='latex inline'>\displaystyle 2 \% </code> each year. This year, each person threw out an average of <code class='latex inline'>\displaystyle 1500 \mathrm{lb} </code> of garbage. The function <code class='latex inline'>\displaystyle y=1500 \cdot 0.98^{x} </code> models the average amount of garbage each person will throw out each year after <code class='latex inline'>\displaystyle x </code> years. Graph the function.</p>
<p>A plastic sun visor allows light to pass through but reduces the intensity of the light. The intensity is reduced by 5% if the plastic is 1 mm thick. Each additional millimetre of thickness reduces the intensity by another 5%.</p><p>a) Use an equation to model the relation between the thickness of the plastic and the intensity of the light.</p><p>b) How thick is a piece of plastic that reduces the intensity of the light to 60%?</p>
<p>For each exponential function, state the base function, <code class='latex inline'>y = 5^x</code>. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.</p><p><code class='latex inline'>\displaystyle y = (\frac{1}{2})^{\frac{x}{2}} -3 </code></p>
<p>For each exponential function, state the base function, <code class='latex inline'>y = 5^x</code>. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph.</p><p><code class='latex inline'>\displaystyle y =-2(3)^{2x +4} </code></p>
<p>a. Graph <code class='latex inline'>\displaystyle y=2^{x}, y=4^{x} </code>, and <code class='latex inline'>\displaystyle y=0.25^{x} </code> on the same axes.</p><p>b. What point is on all three graphs?</p><p>c. Does the graph of an exponential function intersect the <code class='latex inline'>\displaystyle x </code> -axis? Explain.</p><p>d. Reasoning How does the graph of <code class='latex inline'>\displaystyle y=b^{x} </code> change as the base <code class='latex inline'>\displaystyle b </code> increases or decreases?</p>
<p>You have just read a journal article about a population of fungi that doubles every 3 weeks. The beginning population was <code class='latex inline'>\displaystyle 10 . </code> The function <code class='latex inline'>\displaystyle y=10 \cdot 2^{\frac{n}{3}} </code> represents the population after <code class='latex inline'>\displaystyle n </code> weeks.</p><p>a. You have a population of 15 of the same fungi. Assuming the journal articles gives the correct rate of increase, write the function that represents the population of fungi after <code class='latex inline'>\displaystyle n </code> weeks.</p><p>b. Suppose you find another article that states that the fungi population triples every 4 weeks. If there are currently 15 fungi in your population, write the function that represents the population after <code class='latex inline'>\displaystyle n </code> weeks.</p>
<p><code class='latex inline'>\displaystyle \$100 </code> is put into a bank account that pays interest so that the amount in the account grows according to the expression <code class='latex inline'>\displaystyle 100(1.06)^{n} </code>, where <code class='latex inline'>\displaystyle n </code> is the number of years. </p><p>Find the annual interest rate of the account</p> <p> State the transformations that are applied to each parent function, resulting in the given transformed function. Sketch the graphs of the</p><p><code class='latex inline'>f(x) =2^x, y=f(x-1)+2</code></p> <p>Vocabulary Describe the differences between a linear function and an exponential function.</p> <p>State the domain and range of each function.</p><p><code class='latex inline'>f(x) = 10^x</code></p> <img src="/qimages/56531" /><p>Error Analysis A student evaluated</p><p>the function <code class='latex inline'>\displaystyle f(x)=3 \cdot 4^{x} </code> for <code class='latex inline'>\displaystyle x=-1 </code> as shown at the right. Describe and correct the</p><p>student&#39;s mistake.</p> <p>Computers A computer valued at <code class='latex inline'>\displaystyle \$ 1500 </code> loses <code class='latex inline'>\displaystyle 20 \% </code> of its value each year.</p><p>a. Write a function rule that models the value of the computer.</p><p>b. Find the value of the computer after 3 yr.</p><p>c. In how many years will the value of the computer be less than <code class='latex inline'>\displaystyle \\$ 500 ? </code></p>
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