Graphing a Vertical Shift

The first transformation occurs when we add a constant[latex]\,d\,[/latex]to the parent function[latex]\,f\left(x\right)={b}^{x},[/latex] giving us a vertical shift[latex]\,d\,[/latex]units in the same direction as the sign. For example, if we begin by graphing a parent function,[latex]\,f\left(x\right)={2}^{x},[/latex] we can then graph two vertical shifts alongside it, using[latex]\,d=3:\,[/latex]the upward shift,[latex]\,g\left(x\right)={2}^{x}+3\,[/latex]and the downward shift,[latex]\,h\left(x\right)={2}^{x}-3.\,[/latex]Both vertical shifts are shown in (Figure).

Figure 5.

Observe the results of shifting[latex]\,f\left(x\right)={2}^{x}\,[/latex]vertically:

• The domain,[latex]\,\left(-\infty ,\infty \right)\,[/latex]remains unchanged.
• When the function is shifted up[latex]\,3\,[/latex]units to[latex]\,g\left(x\right)={2}^{x}+3:[/latex]
  • The y-intercept shifts up[latex]\,3\,[/latex]units to[latex]\,\left(0,4\right).[/latex]
  • The asymptote shifts up[latex]\,3\,[/latex]units to[latex]\,y=3.[/latex]
  • The range becomes[latex]\,\left(3,\infty \right).[/latex]
• When the function is shifted down[latex]\,3\,[/latex]units to[latex]\,h\left(x\right)={2}^{x}-3:[/latex]
  • The y-intercept shifts down[latex]\,3\,[/latex]units to[latex]\,\left(0,-2\right).[/latex]
  • The asymptote also shifts down[latex]\,3\,[/latex]units to[latex]\,y=-3.[/latex]
  • The range becomes[latex]\,\left(-3,\infty \right).[/latex]

Graphing a Horizontal Shift

The next transformation occurs when we add a constant[latex]\,c\,[/latex]to the input of the parent function[latex]\,f\left(x\right)={b}^{x},[/latex] giving us a horizontal shift[latex]\,c\,[/latex]units in the opposite direction of the sign. For example, if we begin by graphing the parent function[latex]\,f\left(x\right)={2}^{x},[/latex] we can then graph two horizontal shifts alongside it, using[latex]\,c=3:\,[/latex]the shift left,[latex]\,g\left(x\right)={2}^{x+3},[/latex] and the shift right,[latex]\,h\left(x\right)={2}^{x-3}.\,[/latex]Both horizontal shifts are shown in (Figure).

Figure 6.

Observe the results of shifting[latex]\,f\left(x\right)={2}^{x}\,[/latex]horizontally:

• The domain,[latex]\,\left(-\infty ,\infty \right),[/latex]remains unchanged.
• The asymptote,[latex]\,y=0,[/latex]remains unchanged.
• The y-intercept shifts such that:
  • When the function is shifted left[latex]\,3\,[/latex]units to[latex]\,g\left(x\right)={2}^{x+3},[/latex]the y-intercept becomes[latex]\,\left(0,8\right).\,[/latex]This is because[latex]\,{2}^{x+3}=\left(8\right){2}^{x},[/latex]so the initial value of the function is[latex]\,8.[/latex]
  • When the function is shifted right[latex]\,3\,[/latex]units to[latex]\,h\left(x\right)={2}^{x-3},[/latex]the y-intercept becomes[latex]\,\left(0,\frac{1}{8}\right).\,[/latex]Again, see that[latex]\,{2}^{x-3}=\left(\frac{1}{8}\right){2}^{x},[/latex]so the initial value of the function is[latex]\,\frac{1}{8}.[/latex]

Graphing a Shift of an Exponential Function

Graph[latex]\,f\left(x\right)={2}^{x+1}-3.\,[/latex]State the domain, range, and asymptote.

Show Solution

We have an exponential equation of the form[latex]\,f\left(x\right)={b}^{x+c}+d,[/latex] with[latex]\,b=2,[/latex][latex]\,c=1,[/latex] and[latex]\,d=-3.[/latex]

Draw the horizontal asymptote[latex]\,y=d[/latex], so draw[latex]\,y=-3.[/latex]

Identify the shift as[latex]\,\left(-c,d\right),[/latex] so the shift is[latex]\,\left(-1,-3\right).[/latex]

Shift the graph of[latex]\,f\left(x\right)={b}^{x}\,[/latex]left 1 units and down 3 units.

Figure 7.

The domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(-3,\infty \right);\,[/latex]the horizontal asymptote is[latex]\,y=-3.[/latex]

Graphing a Stretch or Compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]by a constant[latex]\,|a|>0.\,[/latex]For example, if we begin by graphing the parent function[latex]\,f\left(x\right)={2}^{x},[/latex]we can then graph the stretch, using[latex]\,a=3,[/latex]to get[latex]\,g\left(x\right)=3{\left(2\right)}^{x}\,[/latex]as shown on the left in (Figure), and the compression, using[latex]\,a=\frac{1}{3},[/latex]to get[latex]\,h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}\,[/latex]as shown on the right in (Figure).

Figure 8. (a)[latex]\,g\left(x\right)=3{\left(2\right)}^{x}\,[/latex]stretches the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]vertically by a factor of[latex]\,3.\,[/latex](b)[latex]\,h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}\,[/latex]compresses the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]vertically by a factor of[latex]\,\frac{1}{3}.[/latex]

Stretches and Compressions of the Parent Function f(x) = b^x

For any factor[latex]\,a>0,[/latex]the function[latex]\,f\left(x\right)=a{\left(b\right)}^{x}[/latex]

• is stretched vertically by a factor of[latex]\,a\,[/latex]if[latex]\,|a|>1.[/latex]
• is compressed vertically by a factor of[latex]\,a\,[/latex]if[latex]\,|a|<1.[/latex]
• has a y-intercept of[latex]\,\left(0,a\right).[/latex]
• has a horizontal asymptote at[latex]\,y=0,[/latex] a range of[latex]\,\left(0,\infty \right),[/latex] and a domain of[latex]\,\left(-\infty ,\infty \right),[/latex]which are unchanged from the parent function.

Graphing the Stretch of an Exponential Function

Sketch a graph of[latex]\,f\left(x\right)=4{\left(\frac{1}{2}\right)}^{x}.\,[/latex]State the domain, range, and asymptote.

Show Solution

Before graphing, identify the behavior and key points on the graph.

• Since[latex]\,b=\frac{1}{2}\,[/latex]is between zero and one, the left tail of the graph will increase without bound as[latex]\,x\,[/latex]decreases, and the right tail will approach the x-axis as[latex]\,x\,[/latex]increases.
• Since[latex]\,a=4,[/latex]the graph of[latex]\,f\left(x\right)={\left(\frac{1}{2}\right)}^{x}\,[/latex]will be stretched by a factor of[latex]\,4.[/latex]
• Create a table of points as shown in (Figure).
  

  [latex]x[/latex]	[latex]-3[/latex]	[latex]-2[/latex]	[latex]-1[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]
  [latex]f\left(x\right)=4\left(\frac{1}{2}\right)^{x}[/latex]	[latex]32[/latex]	[latex]16[/latex]	[latex]8[/latex]	[latex]4[/latex]	[latex]2[/latex]	[latex]1[/latex]	[latex]0.5[/latex]

• Plot the y-intercept,[latex]\,\left(0,4\right),[/latex]along with two other points. We can use[latex]\,\left(-1,8\right)\,[/latex]and[latex]\,\left(1,2\right).[/latex]

Draw a smooth curve connecting the points, as shown in (Figure).

Figure 9.

The domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(0,\infty \right);\,[/latex]the horizontal asymptote is[latex]\,y=0.[/latex]

Try It

Sketch the graph of[latex]\,f\left(x\right)=\frac{1}{2}{\left(4\right)}^{x}.\,[/latex]State the domain, range, and asymptote.

Show Solution

The domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(0,\infty \right);\,[/latex]the horizontal asymptote is[latex]\,y=0.\,[/latex]

Graphing Reflections

In addition to shifting, compressing, and stretching a graph, we can also reflect it about the x-axis or the y-axis. When we multiply the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]by[latex]\,-1,[/latex]we get a reflection about the x-axis. When we multiply the input by[latex]\,-1,[/latex]we get a reflection about the y-axis. For example, if we begin by graphing the parent function[latex]\,f\left(x\right)={2}^{x},[/latex] we can then graph the two reflections alongside it. The reflection about the x-axis,[latex]\,g\left(x\right)={-2}^{x},[/latex]is shown on the left side of (Figure), and the reflection about the y-axis[latex]\,h\left(x\right)={2}^{-x},[/latex] is shown on the right side of (Figure).

Figure 10. (a)[latex]\,g\left(x\right)=-{2}^{x}\,[/latex]reflects the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]about the x-axis. (b)[latex]\,g\left(x\right)={2}^{-x}\,[/latex]reflects the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]about the y-axis.

Reflections of the Parent Function f(x) = b^x

The function[latex]\,f\left(x\right)=-{b}^{x}[/latex]

• reflects the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]about the x-axis.
• has a y-intercept of[latex]\,\left(0,-1\right).[/latex]
• has a range of[latex]\,\left(-\infty ,0\right)[/latex]
• has a horizontal asymptote at[latex]\,y=0\,[/latex]and domain of[latex]\,\left(-\infty ,\infty \right),[/latex]which are unchanged from the parent function.

The function[latex]\,f\left(x\right)={b}^{-x}[/latex]

• reflects the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]about the y-axis.
• has a y-intercept of[latex]\,\left(0,1\right),[/latex] a horizontal asymptote at[latex]\,y=0,[/latex] a range of[latex]\,\left(0,\infty \right),[/latex] and a domain of[latex]\,\left(-\infty ,\infty \right),[/latex] which are unchanged from the parent function.

Writing and Graphing the Reflection of an Exponential Function

Find and graph the equation for a function,[latex]\,g\left(x\right),[/latex]that reflects[latex]\,f\left(x\right)={\left(\frac{1}{4}\right)}^{x}\,[/latex]about the x-axis. State its domain, range, and asymptote.

Show Solution

Since we want to reflect the parent function[latex]\,f\left(x\right)={\left(\frac{1}{4}\right)}^{x}\,[/latex]about the x-axis, we multiply[latex]\,f\left(x\right)\,[/latex]by[latex]\,-1\,[/latex]to get,[latex]\,g\left(x\right)=-{\left(\frac{1}{4}\right)}^{x}.\,[/latex]Next we create a table of points as in (Figure).

[latex]x[/latex]	[latex]-3[/latex]	[latex]-2[/latex]	[latex]-1[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]
[latex]g\left(x\right)=-\left(\frac{1}{4}\right)^{x}[/latex]	[latex]-64[/latex]	[latex]-16[/latex]	[latex]-4[/latex]	[latex]-1[/latex]	[latex]-0.25[/latex]	[latex]-0.0625[/latex]	[latex]-0.0156[/latex]

Plot the y-intercept,[latex]\,\left(0,-1\right),[/latex]along with two other points. We can use[latex]\,\left(-1,-4\right)\,[/latex]and[latex]\,\left(1,-0.25\right).[/latex]

Draw a smooth curve connecting the points:

Figure 11.

The domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(-\infty ,0\right);\,[/latex]the horizontal asymptote is[latex]\,y=0.[/latex]

Try It

Find and graph the equation for a function,[latex]\,g\left(x\right),[/latex] that reflects[latex]\,f\left(x\right)={1.25}^{x}\,[/latex]about the y-axis. State its domain, range, and asymptote.

Show Solution

The domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(0,\infty \right);\,[/latex]the horizontal asymptote is[latex]\,y=0.[/latex]

Summarizing Translations of the Exponential Function

Now that we have worked with each type of translation for the exponential function, we can summarize them in (Figure) to arrive at the general equation for translating exponential functions.

1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b>1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).”>

Translations of the Parent Function [latex]\,f\left(x\right)={b}^{x}[/latex]
Translation	Form
Shift Horizontally[latex]\,c\,[/latex]units to the left Vertically[latex]\,d\,[/latex]units up	[latex]f\left(x\right)={b}^{x+c}+d[/latex]
Stretch and Compress Stretch if[latex]\,|a|>1[/latex] Compression if[latex]\,0<|a|<1[/latex]	[latex]f\left(x\right)=a{b}^{x}[/latex]
Reflect about the x-axis	[latex]f\left(x\right)=-{b}^{x}[/latex]
Reflect about the y-axis	[latex]f\left(x\right)={b}^{-x}={\left(\frac{1}{b}\right)}^{x}[/latex]
General equation for all translations	[latex]f\left(x\right)=a{b}^{x+c}+d[/latex]

Translations of Exponential Functions

A translation of an exponential function has the form

[latex]f\left(x\right)=a{b}^{x+c}+d[/latex]

Where the parent function,[latex]\,y={b}^{x},[/latex][latex]\,b>1,[/latex]is

• shifted horizontally[latex]\,c\,[/latex]units to the left.
• stretched vertically by a factor of[latex]\,|a|\,[/latex]if[latex]\,|a|>0.[/latex]
• compressed vertically by a factor of[latex]\,|a|\,[/latex]if[latex]\,0<|a|<1.[/latex]
• shifted vertically[latex]\,d\,[/latex]units.
• reflected about the x-axis when[latex]\,a<0.[/latex]

Note the order of the shifts, transformations, and reflections follow the order of operations.

Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

• [latex]f\left(x\right)={e}^{x}\,[/latex]is vertically stretched by a factor of[latex]\,2\,[/latex], reflected across the y-axis, and then shifted up[latex]\,4\,[/latex]units.

Show Solution

We want to find an equation of the general form[latex]\,f\left(x\right)=a{b}^{x+c}+d.\,[/latex]We use the description provided to find[latex]\,a,[/latex] [latex]b,[/latex] [latex]c,[/latex] and [latex]\,d.[/latex]

• We are given the parent function[latex]\,f\left(x\right)={e}^{x},[/latex] so[latex]\,b=e.[/latex]
• The function is stretched by a factor of[latex]\,2[/latex], so[latex]\,a=2.[/latex]
• The function is reflected about the y-axis. We replace[latex]\,x\,[/latex]with[latex]\,-x\,[/latex]to get:[latex]\,{e}^{-x}.[/latex]
• The graph is shifted vertically 4 units, so[latex]\,d=4.[/latex]

Substituting in the general form we get,

[latex]\begin{array}{ll} f\left(x\right)\hfill & =a{b}^{x+c}+d\hfill \\ \hfill & =2{e}^{-x+0}+4\hfill \\ \hfill & =2{e}^{-x}+4\hfill \end{array}[/latex]

The domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(4,\infty \right);\,[/latex]the horizontal asymptote is[latex]\,y=4.[/latex]

Try It

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

• [latex]f\left(x\right)={e}^{x}\,[/latex]is compressed vertically by a factor of[latex]\,\frac{1}{3},[/latex] reflected across the x-axis and then shifted down [latex]\,2[/latex] units.

Show Solution

[latex]f\left(x\right)=-\frac{1}{3}{e}^{x}-2;\,[/latex]the domain is[latex]\,\left(-\infty ,\infty \right);\,[/latex]the range is[latex]\,\left(-\infty ,2\right);\,[/latex]the horizontal asymptote is[latex]\,y=2.[/latex]

Access this online resource for additional instruction and practice with graphing exponential functions.

• Graph Exponential Functions

Section Exercises

Verbal

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

Show Solution

An asymptote is a line that the graph of a function approaches, as[latex]\,x\,[/latex]either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

Algebraic

The graph of[latex]\,f\left(x\right)={3}^{x}\,[/latex]is reflected about the y-axis and stretched vertically by a factor of[latex]\,4.\,[/latex]What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range.

Show Solution

[latex]g\left(x\right)=4{\left(3\right)}^{-x};\,[/latex]y-intercept:[latex]\,\left(0,4\right);\,[/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\,0.[/latex]

The graph of[latex]\,f\left(x\right)={\left(\frac{1}{2}\right)}^{-x}\,[/latex]is reflected about the y-axis and compressed vertically by a factor of[latex]\,\frac{1}{5}.\,[/latex]What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range.

The graph of[latex]\,f\left(x\right)={10}^{x}\,[/latex]is reflected about the x-axis and shifted upward[latex]\,7\,[/latex]units. What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range.

Show Solution

[latex]g\left(x\right)=-{10}^{x}+7;\,[/latex]y-intercept:[latex]\,\left(0,6\right);\,[/latex]Domain: all real numbers; Range: all real numbers less than[latex]\,7.[/latex]

The graph of[latex]\,f\left(x\right)={\left(1.68\right)}^{x}\,[/latex]is shifted right[latex]\,3\,[/latex]units, stretched vertically by a factor of[latex]\,2,[/latex]reflected about the x-axis, and then shifted downward[latex]\,3\,[/latex]units. What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept (to the nearest thousandth), domain, and range.

The graph of[latex]\,f\left(x\right)=2{\left(\frac{1}{4}\right)}^{x-20}[/latex] is shifted left[latex]\,2\,[/latex]units, stretched vertically by a factor of[latex]\,4,[/latex]reflected about the x-axis, and then shifted downward[latex]\,4\,[/latex]units. What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range.

Show Solution

[latex]g\left(x\right)=2{\left(\frac{1}{4}\right)}^{x};\,[/latex] y-intercept:[latex]\,\left(0,\text{ 2}\right);\,[/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\,0.[/latex]

Graphical

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

[latex]f\left(x\right)=3{\left(\frac{1}{2}\right)}^{x}[/latex]

[latex]g\left(x\right)=-2{\left(0.25\right)}^{x}[/latex]

Show Solution

y-intercept:[latex]\,\left(0,-2\right)[/latex]

[latex]h\left(x\right)=6{\left(1.75\right)}^{-x}[/latex]

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

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

Show Solution

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

For the following exercises, match each function with one of the graphs in (Figure).

Figure 12.

[latex]f\left(x\right)=2{\left(0.69\right)}^{x}[/latex]

Show Solution

B

[latex]f\left(x\right)=2{\left(1.28\right)}^{x}[/latex]

[latex]f\left(x\right)=2{\left(0.81\right)}^{x}[/latex]

Show Solution

A

[latex]f\left(x\right)=4{\left(1.28\right)}^{x}[/latex]

[latex]f\left(x\right)=2{\left(1.59\right)}^{x}[/latex]

Show Solution

E

[latex]f\left(x\right)=4{\left(0.69\right)}^{x}[/latex]

For the following exercises, use the graphs shown in (Figure). All have the form[latex]\,f\left(x\right)=a{b}^{x}.[/latex]

Figure 13.

Which graph has the largest value for[latex]\,b?[/latex]

Show Solution

D

Which graph has the smallest value for[latex]\,b?[/latex]

Which graph has the largest value for[latex]\,a?[/latex]

Show Solution

C

Which graph has the smallest value for[latex]\,a?[/latex]

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

[latex]f\left(x\right)=\frac{1}{2}{\left(4\right)}^{x}[/latex]

Show Solution

[latex]f\left(x\right)=3{\left(0.75\right)}^{x}-1[/latex]

[latex]f\left(x\right)=-4{\left(2\right)}^{x}+2[/latex]

Show Solution

For the following exercises, graph the transformation of[latex]\,f\left(x\right)={2}^{x}.\,[/latex]Give the horizontal asymptote, the domain, and the range.

[latex]f\left(x\right)={2}^{-x}[/latex]

[latex]h\left(x\right)={2}^{x}+3[/latex]

Show Solution

Horizontal asymptote:[latex]\,h\left(x\right)=3;[/latex] Domain: all real numbers; Range: all real numbers strictly greater than[latex]\,3.[/latex]

[latex]f\left(x\right)={2}^{x-2}[/latex]

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

[latex]f\left(x\right)=-5{\left(4\right)}^{x}-1[/latex]

Show Solution

As [latex]x\to \infty[/latex],
[latex]f\left(x\right)\to -\infty[/latex];

[latex]f\left(x\right)=3{\left(\frac{1}{2}\right)}^{x}-2[/latex]

[latex]f\left(x\right)=3{\left(4\right)}^{-x}+2[/latex]

Show Solution

As [latex]x\to \infty[/latex],
[latex]f\left(x\right)\to 2[/latex];

For the following exercises, start with the graph of[latex]\,f\left(x\right)={4}^{x}.\,[/latex]Then write a function that results from the given transformation.

Shift [latex]f\left(x\right)[/latex] 4 units upward

Shift[latex]\,f\left(x\right)\,[/latex]3 units downward

Show Solution

[latex]f\left(x\right)={4}^{x}-3[/latex]

Shift[latex]\,f\left(x\right)\,[/latex]2 units left

Shift[latex]\,f\left(x\right)\,[/latex]5 units right

Show Solution

[latex]f\left(x\right)={4}^{x-5}[/latex]

Reflect[latex]\,f\left(x\right)\,[/latex]about the x-axis

Reflect[latex]\,f\left(x\right)\,[/latex]about the y-axis

Show Solution

[latex]f\left(x\right)={4}^{-x}[/latex]

For the following exercises, each graph is a transformation of[latex]\,y={2}^{x}.\,[/latex]Write an equation describing the transformation.

Show Solution

[latex]y=-{2}^{x}+3[/latex]

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

Show Solution

[latex]y=-2{\left(3\right)}^{x}+7[/latex]

Numeric

For the following exercises, evaluate the exponential functions for the indicated value of[latex]\,x.[/latex]

[latex]g\left(x\right)=\frac{1}{3}{\left(7\right)}^{x-2}\,[/latex]for[latex]\,g\left(6\right).[/latex]

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[latex]g\left(6\right)=800+\frac{1}{3}\approx 800.3333[/latex]

[latex]f\left(x\right)=4{\left(2\right)}^{x-1}-2\,[/latex]for[latex]\,f\left(5\right).[/latex]

[latex]h\left(x\right)=-\frac{1}{2}{\left(\frac{1}{2}\right)}^{x}+6\,[/latex]for[latex]\,h\left(-7\right).[/latex]

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[latex]h\left(-7\right)=-58[/latex]

Technology

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

[latex]-50=-{\left(\frac{1}{2}\right)}^{-x}[/latex]

[latex]116=\frac{1}{4}{\left(\frac{1}{8}\right)}^{x}[/latex]

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[latex]x\approx -2.953[/latex]

[latex]12=2{\left(3\right)}^{x}+1[/latex]

[latex]5=3{\left(\frac{1}{2}\right)}^{x-1}-2[/latex]

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[latex]x\approx -0.222[/latex]

[latex]-30=-4{\left(2\right)}^{x+2}+2[/latex]

Extensions

Explore and discuss the graphs of[latex]\,F\left(x\right)={\left(b\right)}^{x}\,[/latex]and[latex]\,G\left(x\right)={\left(\frac{1}{b}\right)}^{x}.\,[/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\,{b}^{x}\,[/latex]and[latex]\,{\left(\frac{1}{b}\right)}^{x}\,[/latex]for any real number[latex]\,b>0.[/latex]

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The graph of[latex]\,G\left(x\right)={\left(\frac{1}{b}\right)}^{x}\,[/latex]is the refelction about the y-axis of the graph of[latex]\,F\left(x\right)={b}^{x};\,[/latex]For any real number[latex]\,b>0\,[/latex]and function[latex]\,f\left(x\right)={b}^{x},[/latex]the graph of[latex]\,{\left(\frac{1}{b}\right)}^{x}\,[/latex]is the the reflection about the y-axis,[latex]\,F\left(-x\right).[/latex]

Prove the conjecture made in the previous exercise.

Explore and discuss the graphs of[latex]\,f\left(x\right)={4}^{x},[/latex][latex]\,g\left(x\right)={4}^{x-2},[/latex]and[latex]\,h\left(x\right)=\left(\frac{1}{16}\right){4}^{x}.\,[/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\,{b}^{x}\,[/latex]and[latex]\,\left(\frac{1}{{b}^{n}}\right){b}^{x}\,[/latex]for any real number n and real number[latex]\,b>0.[/latex]

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The graphs of[latex]\,g\left(x\right)\,[/latex]and[latex]\,h\left(x\right)\,[/latex]are the same and are a horizontal shift to the right of the graph of[latex]\,f\left(x\right);\,[/latex]For any real number n, real number[latex]\,b>0,[/latex] and function[latex]\,f\left(x\right)={b}^{x},[/latex] the graph of[latex]\,\left(\frac{1}{{b}^{n}}\right){b}^{x}\,[/latex]is the horizontal shift[latex]\,f\left(x-n\right).[/latex]

Prove the conjecture made in the previous exercise.