13.2.2: Homework

Math 375: Pre-Calculus · LibreTexts

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    Reading Questions

    1. What is the "constant ratio" for an exponential function of the form \(f(x)=ab^x\), and what does it signify about the function's values as the input increases by 1?
    2. Describe the domain, range, and horizontal asymptote of the basic exponential function \(f(x)=b^x\) where \(b>0\) and \(b \neq 1\).
    3. How does a vertical shift by a constant \(d\) (i.e., \(g(x)=b^x+d\)) affect the domain, range, and horizontal asymptote of \(f(x)=b^x\)?
    4. How does a horizontal shift by a constant \(c\) (i.e., \(g(x)=b^{x+c}\)) affect the domain, range, and horizontal asymptote of \(f(x)=b^x\)?
    5. Explain how the graph of \(g(x) = a \cdot b^x\) is related to the graph of \(f(x)=b^x\) if \(|a|>1\). What if \(0 < |a| < 1\)?
    6. How is the graph of \(g(x)=-b^x\) related to the graph of \(f(x)=b^x\)?
    7. How is the graph of \(g(x)=b^{-x}\) related to the graph of \(f(x)=b^x\)?
    8. When graphing \(f(x) = ab^{x+c}+d\), what is a general order in which transformations can be applied to the parent graph \(y=b^x\)?
    9. If the graph of \(f(x)=e^x\) is vertically stretched by a factor of 2, reflected across the \(y\)-axis, and then shifted up 4 units, what is the equation of the new function?
    10. What is the \(y\)-intercept of the function \(f(x) = 4(\frac{1}{2})^x\)?
    11. What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?
    12. What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

    Homework

    Algebraic

    1. The graph of \(f(x)= 3^x\) is reflected about the \( y \)-axis and stretched vertically by a factor of \(4\). What is the equation of the new function, \(g(x)\)? State its \( y \)-intercept, domain, and range.
    2. The graph of \(f(x)= \left( \frac{1}{2} \right)^{-x}\) is reflected about the \( y \)-axis and compressed vertically by a factor of \(\frac{1}{5} \). What is the equation of the new function, \(g(x)\)? State its \( y \)-intercept, domain, and range.
    3. The graph of \(f(x)= 10^x\) is reflected about the \( x \)-axis and shifted upward \(7\) units. What is the equation of the new function, \(g(x)\)? State its \( y \)-intercept, domain, and range.
    4. The graph of \(f(x)= ( 1.68 )^x\) is shifted right \(3\) units, stretched vertically by a factor of \(2\), reflected about the \( x \)-axis, and then shifted downward \(3\) units. What is the equation of the new function, \(g(x)\)? State its \( y \)-intercept (to the nearest thousandth), domain, and range.
    5. The graph of \(f(x)=-\frac{1}{2}\left(\frac{1}{4}\right)^{x-2}+4\) is shifted downward \(4\) units, and then shifted left \(2\) units, stretched vertically by a factor of \(4\), and reflected about the \( x \)-axis. What is the equation of the new function, \(g(x)\)? State its \( y \)-intercept, domain, and range.

    Graphical

    For the following exercises, graph the function and its reflection about the \( y \)-axis on the same axes, and give the \( y \)-intercept.

    1. \(f(x)=3 \left( \frac{1}{2} \right)^x\)
    2. \(g(x)=−2 ( 0.25 )^x\)
    3. \(h(x)=6 ( 1.75 )^{-x}\)

    For the following exercises, graph each set of functions on the same axes.

    1. \(f(x)=3 \left( \frac{1}{4} \right)^x \), \(g(x)=3 ( 2 )^x \), and \(h(x)=3 ( 4 )^x\)
    2. \(f(x)= \frac{1}{4} ( 3 )^x \), \(g(x)=2 ( 3 )^x \), and \(h(x)=4 ( 3 )^x\)

    For the following exercises, match each function with one of the graphs in Figure \( \PageIndex{ 1 } \).

    Figure \( \PageIndex{ 1 } \)
     
    Graph of six exponential functions.
    1. \(f( x )=2 ( 0.69 )^x\)
    2. \(f( x )=2 ( 1.28 )^x\)
    3. \(f( x )=2 ( 0.81 )^x\)
    4. \(f( x )=4 ( 1.28 )^x\)
    5. \(f( x )=2 ( 1.59 )^x\)
    6. \(f( x )=4 ( 0.69 )^x\)

    For the following exercises, use the graphs shown in Figure \( \PageIndex{ 2 } \). All have the form \(f( x )= a b^x \).

    Figure \( \PageIndex{ 2 } \)
     
    Graph of six exponential functions.
    1. Which graph has the largest value for \(b\)?
    2. Which graph has the smallest value for \(b\)?
    3. Which graph has the largest value for \(a\)?
    4. Which graph has the smallest value for \(a\)?

    For the following exercises, graph the function and its reflection about the \( x \)-axis on the same axes.

    1. \(f(x)= \frac{1}{2} ( 4 )^x\)
    2. \(f(x)=3 ( 0.75 )^x −1\)
    3. \(f(x)=−4 ( 2 )^x +2\)

    For the following exercises, graph the transformation of \(f(x)= 2^x \). Give the horizontal asymptote, the domain, and the range.

    1. \(f( x )= 2^{−x}\)
    2. \(h( x )= 2^x +3\)
    3. \(f( x )= 2^{x−2}\)

    For the following exercises, describe the end behavior of the graphs of the functions.

    1. \(f( x )=−5 ( 4 )^x −1\)
    2. \(f( x )=3 \left( \frac{1}{2} \right)^x −2\)
    3. \(f( x )=3 ( 4 )^{-x} +2\)

    For the following exercises, start with the graph of \(f( x )= 4^x \). Then write a function that results from the given transformation.

    1. Shift \(f(x)\) 4 units upward
    2. Shift \(f(x)\) 3 units downward
    3. Shift \(f(x)\) 2 units left
    4. Shift \(f(x)\) 5 units right
    5. Reflect \(f(x)\) about the \( x \)-axis
    6. Reflect \(f(x)\) about the \( y \)-axis

    For the following exercises, each graph is a transformation of \(y= 2^x \). Write an equation describing the transformation.

    1.  
      Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.
    2.  
      Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.
    3.  
      Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.

    For the following exercises, find an exponential equation for the graph.

    1.  
      Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.
    2.  
      Graph of f(x)=(1/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.

    Numeric

    For the following exercises, evaluate the exponential functions for the indicated value of \(x\).

    1. \(g(x)= \frac{1}{3} ( 7 )^{x−2}\) for \(g(6)\).
    2. \(f(x)=4 (2)^{x−1} −2\) for \(f(5)\).
    3. \(h(x)=− \frac{1}{2} \left( \frac{1}{2} \right)^x +6\) for \(h(−7)\).

    Technology

    For the following exercises, use graphing technology to approximate the solutions of the equation. Round to the nearest thousandth.

    1. \(−50=− \left( \frac{1}{2} \right)^{-x}\)
    2. \(116= \frac{1}{4} \left( \frac{1}{8} \right)^x\)
    3. \(12=2 ( 3 )^x +1\)
    4. \(5=3 \left( \frac{1}{2} \right)^{x−1} −2\)
    5. \(−30=−4 ( 2 )^{x+2} +2\)

    Extensions

    1. Explore and discuss the graphs of \(F(x)= b^x\) and \(G(x)= \left( \frac{1}{b} \right)^x \). Then make a conjecture about the relationship between the graphs of the functions \(b^x\) and \(\left( \frac{1}{b} \right)^x\) for any real number \(b>0\).
    2. Prove the conjecture made in the previous exercise.
    3. Explore and discuss the graphs of \(f(x)= 4^x \), \(g(x)= 4^{x−2} \), and \(h(x)=\left( \frac{1}{16} \right) 4^x \). Then make a conjecture about the relationship between the graphs of the functions \(b^x\) and \(\left( \frac{1}{b^n} \right) b^x\) for any real number \( n \) and real number \(b>0\).
    4. Prove the conjecture made in the previous exercise.

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