6.4 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 $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

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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