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Given a logarithmic function with the form $\text{\hspace{0.17em}}f(x)=a{\mathrm{log}}_{b}\left(x\right),$ $a>0,$ graph the translation.
Sketch a graph of $\text{\hspace{0.17em}}f(x)=2{\mathrm{log}}_{4}(x)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Since the function is $\text{\hspace{0.17em}}f(x)=2{\mathrm{log}}_{4}(x),$ we will notice $\text{\hspace{0.17em}}a=2.$
This means we will stretch the function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{4}(x)\text{\hspace{0.17em}}$ by a factor of 2.
The vertical asymptote is $\text{\hspace{0.17em}}x=0.$
Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{4},\mathrm{-1}\right),$ $(1,0),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,1\right).$
The new coordinates are found by multiplying the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates by 2.
Label the points $\text{\hspace{0.17em}}\left(\frac{1}{4},\mathrm{-2}\right),$ $\left(1,0\right)\text{\hspace{0.17em}},$ and $\text{\hspace{0.17em}}\left(4,\text{2}\right).$
The domain is $\text{\hspace{0.17em}}\left(\mathrm{0,}\text{\hspace{0.17em}}\infty \right),$ the range is $\text{\hspace{0.17em}}(-\infty ,\infty ),\text{\hspace{0.17em}}$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ See [link] .
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 $\text{\hspace{0.17em}}f(x)=\frac{1}{2}\text{\hspace{0.17em}}{\mathrm{log}}_{4}(x)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x=0.$
Sketch a graph of $\text{\hspace{0.17em}}f(x)=5\mathrm{log}(x+2).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link] . The vertical asymptote will be shifted to $\text{\hspace{0.17em}}x=\mathrm{-2.}\text{\hspace{0.17em}}$ The x -intercept will be $\text{\hspace{0.17em}}(\mathrm{-1,}0).\text{\hspace{0.17em}}$ The domain will be $\text{\hspace{0.17em}}\left(\mathrm{-2},\infty \right).\text{\hspace{0.17em}}$ Two points will help give the shape of the graph: $\text{\hspace{0.17em}}(\mathrm{-1},0)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(8,5).\text{\hspace{0.17em}}$ We chose $\text{\hspace{0.17em}}x=8\text{\hspace{0.17em}}$ as the x -coordinate of one point to graph because when $\text{\hspace{0.17em}}x=\mathrm{8,}\text{\hspace{0.17em}}$ $\text{\hspace{0.17em}}x+2=\mathrm{10,}\text{\hspace{0.17em}}$ the base of the common logarithm.
The domain is $\text{\hspace{0.17em}}\left(-2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=-2.$
Sketch a graph of the function $\text{\hspace{0.17em}}f(x)=3\mathrm{log}(x-2)+1.\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
The domain is $\text{\hspace{0.17em}}\left(2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=2.$
When the parent function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is multiplied by $\text{\hspace{0.17em}}\mathrm{-1},$ the result is a reflection about the x -axis. When the input is multiplied by $\text{\hspace{0.17em}}\mathrm{-1},$ the result is a reflection about the y -axis. To visualize reflections, we restrict $\text{\hspace{0.17em}}b>\mathrm{1,}\text{\hspace{0.17em}}$ 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 reflection about the x -axis, $\text{\hspace{0.17em}}g(x)={\mathrm{-log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and the reflection about the y -axis, $\text{\hspace{0.17em}}h(x)={\mathrm{log}}_{b}\left(-x\right).$
The function $\text{\hspace{0.17em}}f(x)={\mathrm{-log}}_{b}\left(x\right)$
The function
$\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(-x\right)$
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