5) 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.
6) 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.
Graphical
For the following exercises, graph the function and its reflection across the \(y\)-axis on the same axes, and give the \(y\)-intercept.
8) \(f(x)=3\left ( \frac{1}{2} \right )^x\)
Exercise \(\PageIndex{9}\)
\(g(x)=-2(0.25)^x\)
- Answer
-
\(y\)-intercept: \((0,-2)\)
10) \(h(x)=6(1.75)^{-x}\)
For the following exercises, graph each set of functions on the same axes.
11) \(f(x)=3\left ( \frac{1}{4} \right )^x, g(x)=3(2)^x, h(x)=3(4)^x\)
- Answer:
-
12) \(f(x)=\frac{1}{4}(3)^x, g(x)=2(3)^x, h(x)=4(3)^x\)
For the following exercises, match each function with one of the graphs in Figure below.
13) \(f(x)=2(0.69)^x\)
- Answer:
-
B
14) \(f(x)=2(1.28)^x\)
15) \(f(x)=2(0.81)^x\)
- Answer:
-
A
16) \(f(x)=4(1.28)^x\)
17) \(f(x)=2(1.59)^x\)
- Answer:
-
E
18) \(f(x)=4(0.69)^x\)
For the following exercises, use the graphs shown in Figure below. All have the form \(f(x)=ab^x\).
Exercise \(\PageIndex{19}\)
Which graph has the largest value for \(b\) ?
- Answer
-
D
Exercise \(\PageIndex{20}\)
Which graph has the smallest value for \(b\)?
Exercise \(\PageIndex{21}\)
Which graph has the largest value for \(a\)?
- Answer
-
C
Exercise \(\PageIndex{22}\)
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.
23) \(f(x)=\frac{1}{2}(4)^x\)
- Answer:
-
24) \(f(x)=3(0.75)^x-1\)
25) \(f(x)=-4(2)^x+2\)
- Answer:
-
For the following exercises, graph the transformation of \(f(x)=2^x\) Give the horizontal asymptote, the domain, and the range.
26) \(f(x)=2^{-x}\)
Exercise \(\PageIndex{27}\)
\(h(x)=2^x+3\)
- Answer
-
Horizontal asymptote: \(y=3\) Domain: all real numbers; Range: all real numbers strictly greater than \(3\).
28) \(f(x)=2^{x-2}\)
For the following exercises, describe the end behavior of the graphs of the functions.
29) \(f(x)=-5(4)^x-1\)
- Answer:
-
As \(x\rightarrow \infty , f(x)\rightarrow -\infty\)
As \(x\rightarrow -\infty , f(x)\rightarrow -1\)
30) \(f(x)=3\left ( \frac{1}{2} \right )^x-2\)
31) \(f(x)=3(4)^{-x}+2\)
- Answer:
-
As \(x\rightarrow \infty , f(x)\rightarrow 2\)
As \(x\rightarrow -\infty , f(x)\rightarrow \infty\)
For the following exercises, start with the graph of \(f(x)=4^x\) Then write a function that results from the given transformation.
32) Shift \(f(x)\) \(4\) units upward
33) Shift \(f(x)\) \(3\) units downward
- Answer:
-
\(f(x)=4^x-3\)
34) Shift \(f(x)\) \(2\) units left
35) Shift \(f(x)\) \(5\) units right
- Answer:
-
\(f(x)=4^{x-5}\)
36) Reflect \(f(x)\) about the \(x\)-axis
37) Reflect \(f(x)\) about the \(y\)-axis
- Answer:
-
\(f(x)=4^{-x}\)
For the following exercises, each graph is a transformation of \(f(x)=2^x\) Write an equation describing the transformation.
38)
Exercise \(\PageIndex{39}\)
- Answer
-
\(y=-2^x+3\)
40)
For the following exercises, find an exponential equation for the graph.
Exercise \(\PageIndex{41}\)
- Answer
-
\(y=-2(3)^x+7\)
42)
Technology
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.
46) \(-50=\left ( \frac{1}{2} \right )^{-x}\)
47) \(116=\left ( \frac{1}{4} \right )\left ( \frac{1}{8} \right )^x\)
- Answer:
-
\(x\approx -2.953\)
48) \(12=2(3)^x+1\)
49) \(5=3\left ( \frac{1}{2} \right )^{x-1}-2\)
- Answer:
-
\(x\approx -0.222\)
50) \(-30=-4(2)^{x+2}+2\)
Extensions
51) 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 \(b>0\).
- Answer:
-
The graph of \(G(x)=\left ( \frac{1}{b} \right )^x\) is the refelction about the \(y\)-axis of the graph of \(F(x)=(b)^x\); For any real number \(b>0\) and function \(f(x)=(b)^x\) the graph of \(\left ( \frac{1}{b} \right )^x\) is the the reflection about the \(y\)-axis, \(F(-x)\).
52) Prove the conjecture made in the previous exercise.
53) 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 \(b>0\).
- Answer:
-
The graphs of \(g(x)\) and \(h(x)\) are the same and are a horizontal shift to the right of the graph of \(f(x)\); For any real number \(n\), real number \(b>0\), and function \(f(x)=b^x\) the graph of \(\left ( \frac{1}{b^n} \right )b^x\) is the horizontal shift \(f(x-n)\).
54) Prove the conjecture made in the previous exercise.