# 6.2 Graphs of exponential functions  (Page 5/6)

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Find and graph the equation for a function, $\text{\hspace{0.17em}}g\left(x\right),$ that reflects $\text{\hspace{0.17em}}f\left(x\right)={1.25}^{x}\text{\hspace{0.17em}}$ about the y -axis. State its domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(0,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=0.$

## 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 [link] to arrive at the general equation for translating exponential functions.

Translations of the Parent Function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}$
Translation Form
Shift
• Horizontally $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left
• Vertically $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units up
$f\left(x\right)={b}^{x+c}+d$
Stretch and Compress
• Stretch if $\text{\hspace{0.17em}}|a|>1$
• Compression if $\text{\hspace{0.17em}}0<|a|<1$
$f\left(x\right)=a{b}^{x}$
Reflect about the x -axis $f\left(x\right)=-{b}^{x}$
Reflect about the y -axis $f\left(x\right)={b}^{-x}={\left(\frac{1}{b}\right)}^{x}$
General equation for all translations $f\left(x\right)=a{b}^{x+c}+d$

## Translations of exponential functions

A translation of an exponential function has the form

Where the parent function, $\text{\hspace{0.17em}}y={b}^{x},$ $\text{\hspace{0.17em}}b>1,$ is

• shifted horizontally $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left.
• stretched vertically by a factor of $\text{\hspace{0.17em}}|a|\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}|a|>0.$
• compressed vertically by a factor of $\text{\hspace{0.17em}}|a|\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0<|a|<1.$
• shifted vertically $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units.
• reflected about the x- axis when $\text{\hspace{0.17em}}a<0.$

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.

• $f\left(x\right)={e}^{x}\text{\hspace{0.17em}}$ is vertically stretched by a factor of $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ , reflected across the y -axis, and then shifted up $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ units.

We want to find an equation of the general form We use the description provided to find $\text{\hspace{0.17em}}a,$ $b,$ $c,$ and $\text{\hspace{0.17em}}d.$

• We are given the parent function $\text{\hspace{0.17em}}f\left(x\right)={e}^{x},$ so $\text{\hspace{0.17em}}b=e.$
• The function is stretched by a factor of $\text{\hspace{0.17em}}2$ , so $\text{\hspace{0.17em}}a=2.$
• The function is reflected about the y -axis. We replace $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}-x\text{\hspace{0.17em}}$ to get: $\text{\hspace{0.17em}}{e}^{-x}.$
• The graph is shifted vertically 4 units, so $\text{\hspace{0.17em}}d=4.$

Substituting in the general form we get,

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(4,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=4.$

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

• $f\left(x\right)={e}^{x}\text{\hspace{0.17em}}$ is compressed vertically by a factor of $\text{\hspace{0.17em}}\frac{1}{3},$ reflected across the x -axis and then shifted down $\text{\hspace{0.17em}}2$ units.

$f\left(x\right)=-\frac{1}{3}{e}^{x}-2;\text{\hspace{0.17em}}$ the domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(-\infty ,2\right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=2.$

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

## Key equations

 General Form for the Translation of the Parent Function $f\left(x\right)=a{b}^{x+c}+d$

## Key concepts

• The graph of the function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ has a y- intercept at domain range and horizontal asymptote $\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See [link] .
• If $\text{\hspace{0.17em}}b>1,$ the function is increasing. The left tail of the graph will approach the asymptote $\text{\hspace{0.17em}}y=0,$ and the right tail will increase without bound.
• If $\text{\hspace{0.17em}}0 the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $\text{\hspace{0.17em}}y=0.$
• The equation $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}+d\text{\hspace{0.17em}}$ represents a vertical shift of the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}.$
• The equation $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}\text{\hspace{0.17em}}$ represents a horizontal shift of the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}.\text{\hspace{0.17em}}$ See [link] .
• Approximate solutions of the equation $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}+d\text{\hspace{0.17em}}$ can be found using a graphing calculator. See [link] .
• The equation $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x},$ where $\text{\hspace{0.17em}}a>0,$ represents a vertical stretch if $\text{\hspace{0.17em}}|a|>1\text{\hspace{0.17em}}$ or compression if $\text{\hspace{0.17em}}0<|a|<1\text{\hspace{0.17em}}$ of the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}.\text{\hspace{0.17em}}$ See [link] .
• When the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ is multiplied by $\text{\hspace{0.17em}}-1,$ the result, $\text{\hspace{0.17em}}f\left(x\right)=-{b}^{x},$ is a reflection about the x -axis. When the input is multiplied by $\text{\hspace{0.17em}}-1,$ the result, $\text{\hspace{0.17em}}f\left(x\right)={b}^{-x},$ is a reflection about the y -axis. See [link] .
• All translations of the exponential function can be summarized by the general equation $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x+c}+d.\text{\hspace{0.17em}}$ See [link] .
• Using the general equation $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x+c}+d,$ we can write the equation of a function given its description. See [link] .

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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