4.4 Graphs of logarithmic functions  (Page 4/8)

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Sketch a graph of $f(x)={\log }_3(x+4)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

The domain is $\left(-4,\infty \right),$ the range $\left(-\infty ,\infty \right),$ and the asymptote $x=–4.$

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Graphing a vertical shift of y = log _b ( x )

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

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

Sketch a graph of $f(x)={\log }_3(x)-2$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is $f(x)={\log }_3(x)-2,$ we will notice $d=–2.$ Thus $d<0.$

This means we will shift the function $f(x)={\log }_3(x)$ down 2 units.

The vertical asymptote is $x=0.$

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

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

Label the points $\left(\frac{1}{3},\mathrm{-3}\right),$ $\left(1,\mathrm{-2}\right),$ and $\left(3,\mathrm{-1}\right).$

The domain is $\left(0,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=0.$

The domain is $\left(0,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=0.$

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Sketch a graph of $f(x)={\log }_2(x)+2$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

The domain is $\left(0,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=0.$

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Graphing stretches and compressions of y = log _b ( x )

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