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Sketch a graph of $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x+4)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x=\u20134.$
When a constant $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ is added to the parent function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right),$ 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 $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ alongside the shift up, $\text{\hspace{0.17em}}g(x)={\mathrm{log}}_{b}\left(x\right)+d\text{\hspace{0.17em}}$ and the shift down, $\text{\hspace{0.17em}}h(x)={\mathrm{log}}_{b}\left(x\right)-d.$ See [link] .
For any constant $\text{\hspace{0.17em}}d,$ the function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)+d$
Given a logarithmic function with the form $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)+d,$ graph the translation.
Sketch a graph of $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x)-2\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Since the function is $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x)-2,$ we will notice $\text{\hspace{0.17em}}d=\u20132.\text{\hspace{0.17em}}$ Thus $\text{\hspace{0.17em}}d<0.$
This means we will shift the function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x)\text{\hspace{0.17em}}$ down 2 units.
The vertical asymptote is $\text{\hspace{0.17em}}x=0.$
Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{3},\mathrm{-1}\right),$ $\left(1,0\right),$ and $\text{\hspace{0.17em}}\left(3,1\right).$
The new coordinates are found by subtracting 2 from the y coordinates.
Label the points $\text{\hspace{0.17em}}\left(\frac{1}{3},\mathrm{-3}\right),$ $\left(1,\mathrm{-2}\right),$ and $\text{\hspace{0.17em}}\left(3,\mathrm{-1}\right).$
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 $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{2}(x)+2\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x=0.$
When the parent function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ alongside the vertical stretch, $\text{\hspace{0.17em}}g(x)=a{\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and the vertical compression, $\text{\hspace{0.17em}}h(x)=\frac{1}{a}{\mathrm{log}}_{b}\left(x\right).$ See [link] .
For any constant $\text{\hspace{0.17em}}a>1,$ the function $\text{\hspace{0.17em}}f(x)=a{\mathrm{log}}_{b}\left(x\right)$
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