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

 Page 4 / 8

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

bsc F. y algebra and trigonometry pepper 2
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4
DB
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-16a+6b+2c
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Joeval
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sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
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x²-2x+9-4x²+12x-20 -3x²+10x+11
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master
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master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
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explain and give four example of hyperbolic function
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y/y+10
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A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
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A banana.
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given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
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Miranda
I am living in india
jai
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Miranda
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I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
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Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Miranda
Jeffrey By By By Heather McAvoy By Stephen Voron By Brooke Delaney By Frank Levy By Stephen Voron By OpenStax By Madison Christian By Vanessa Soledad By Michael Nelson By Richley Crapo