# 4.4 Graphs of logarithmic functions  (Page 4/8)

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Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x+4\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

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

## Graphing a vertical shift of y = log b ( x )

When a constant $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ is added to the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right),$ the result is a vertical shift     $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units in the direction of the sign on $\text{\hspace{0.17em}}d.\text{\hspace{0.17em}}$ To visualize vertical shifts, we can observe the general graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ alongside the shift up, $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d\text{\hspace{0.17em}}$ and the shift down, $\text{\hspace{0.17em}}h\left(x\right)={\mathrm{log}}_{b}\left(x\right)-d.$ See [link] .

## Vertical shifts of the parent function y = log b ( x )

For any constant $\text{\hspace{0.17em}}d,$ the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d$

• shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ up $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}d>0.$
• shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ down $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}d<0.$
• has the vertical asymptote $\text{\hspace{0.17em}}x=0.$
• has domain $\text{\hspace{0.17em}}\left(0,\infty \right).$
• has range $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

Given a logarithmic function with the form $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d,$ graph the translation.

1. Identify the vertical shift:
• If $\text{\hspace{0.17em}}d>0,$ shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ up $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units.
• If $\text{\hspace{0.17em}}d<0,$ shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ down $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units.
2. Draw the vertical asymptote $\text{\hspace{0.17em}}x=0.$
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by adding $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ to the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinate.
4. Label the three points.
5. The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

## Graphing a vertical shift of the parent function y = log b ( x )

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2,$ we will notice $\text{\hspace{0.17em}}d=–2.\text{\hspace{0.17em}}$ Thus $\text{\hspace{0.17em}}d<0.$

This means we will shift the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x\right)\text{\hspace{0.17em}}$ down 2 units.

The vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{3},-1\right),$ $\left(1,0\right),$ and $\text{\hspace{0.17em}}\left(3,1\right).$

The new coordinates are found by subtracting 2 from the y coordinates.

Label the points $\text{\hspace{0.17em}}\left(\frac{1}{3},-3\right),$ $\left(1,-2\right),$ and $\text{\hspace{0.17em}}\left(3,-1\right).$

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

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

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(x\right)+2\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

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

## Graphing stretches and compressions of y = log b ( x )

When the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is multiplied by a constant $\text{\hspace{0.17em}}a>0,$ the result is a vertical stretch    or compression of the original graph. To visualize stretches and compressions, we set $\text{\hspace{0.17em}}a>1\text{\hspace{0.17em}}$ and observe the general graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ alongside the vertical stretch, $\text{\hspace{0.17em}}g\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and the vertical compression, $\text{\hspace{0.17em}}h\left(x\right)=\frac{1}{a}{\mathrm{log}}_{b}\left(x\right).$ See [link] .

## Vertical stretches and compressions of the parent function y = log b ( x )

For any constant $\text{\hspace{0.17em}}a>1,$ the function $\text{\hspace{0.17em}}f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)$

• stretches the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ vertically by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}a>1.$
• compresses the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ vertically by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0
• has the vertical asymptote $\text{\hspace{0.17em}}x=0.$
• has the x -intercept $\text{\hspace{0.17em}}\left(1,0\right).$
• has domain $\text{\hspace{0.17em}}\left(0,\infty \right).$
• has range $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

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