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

 Page 5 / 6

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] .

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