# 6.2 Graphs of exponential functions  (Page 5/6)

 Page 5 / 6

Find and graph the equation for a function, $\text{\hspace{0.17em}}g\left(x\right),$ that reflects $\text{\hspace{0.17em}}f\left(x\right)={1.25}^{x}\text{\hspace{0.17em}}$ about the y -axis. State its domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(0,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=0.$

## Summarizing translations of the exponential function

Now that we have worked with each type of translation for the exponential function, we can summarize them in [link] to arrive at the general equation for translating exponential functions.

Translations of the Parent Function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}$
Translation Form
Shift
• Horizontally $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left
• Vertically $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units up
$f\left(x\right)={b}^{x+c}+d$
Stretch and Compress
• Stretch if $\text{\hspace{0.17em}}|a|>1$
• Compression if $\text{\hspace{0.17em}}0<|a|<1$
$f\left(x\right)=a{b}^{x}$
Reflect about the x -axis $f\left(x\right)=-{b}^{x}$
Reflect about the y -axis $f\left(x\right)={b}^{-x}={\left(\frac{1}{b}\right)}^{x}$
General equation for all translations $f\left(x\right)=a{b}^{x+c}+d$

## Translations of exponential functions

A translation of an exponential function has the form

Where the parent function, $\text{\hspace{0.17em}}y={b}^{x},$ $\text{\hspace{0.17em}}b>1,$ is

• shifted horizontally $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left.
• stretched vertically by a factor of $\text{\hspace{0.17em}}|a|\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}|a|>0.$
• compressed vertically by a factor of $\text{\hspace{0.17em}}|a|\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0<|a|<1.$
• shifted vertically $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units.
• reflected about the x- axis when $\text{\hspace{0.17em}}a<0.$

Note the order of the shifts, transformations, and reflections follow the order of operations.

## Writing a function from a description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

• $f\left(x\right)={e}^{x}\text{\hspace{0.17em}}$ is vertically stretched by a factor of $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ , reflected across the y -axis, and then shifted up $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ units.

We want to find an equation of the general form We use the description provided to find $\text{\hspace{0.17em}}a,$ $b,$ $c,$ and $\text{\hspace{0.17em}}d.$

• We are given the parent function $\text{\hspace{0.17em}}f\left(x\right)={e}^{x},$ so $\text{\hspace{0.17em}}b=e.$
• The function is stretched by a factor of $\text{\hspace{0.17em}}2$ , so $\text{\hspace{0.17em}}a=2.$
• The function is reflected about the y -axis. We replace $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}-x\text{\hspace{0.17em}}$ to get: $\text{\hspace{0.17em}}{e}^{-x}.$
• The graph is shifted vertically 4 units, so $\text{\hspace{0.17em}}d=4.$

Substituting in the general form we get,

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(4,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=4.$

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

• $f\left(x\right)={e}^{x}\text{\hspace{0.17em}}$ is compressed vertically by a factor of $\text{\hspace{0.17em}}\frac{1}{3},$ reflected across the x -axis and then shifted down $\text{\hspace{0.17em}}2$ units.

$f\left(x\right)=-\frac{1}{3}{e}^{x}-2;\text{\hspace{0.17em}}$ the domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(-\infty ,2\right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=2.$

Access this online resource for additional instruction and practice with graphing exponential functions.

## Key equations

 General Form for the Translation of the Parent Function $f\left(x\right)=a{b}^{x+c}+d$

## Key concepts

• The graph of the function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ has a y- intercept at domain range and horizontal asymptote $\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See [link] .
• If $\text{\hspace{0.17em}}b>1,$ the function is increasing. The left tail of the graph will approach the asymptote $\text{\hspace{0.17em}}y=0,$ and the right tail will increase without bound.
• If $\text{\hspace{0.17em}}0 the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $\text{\hspace{0.17em}}y=0.$
• The equation $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}+d\text{\hspace{0.17em}}$ represents a vertical shift of the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}.$
• The equation $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}\text{\hspace{0.17em}}$ represents a horizontal shift of the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}.\text{\hspace{0.17em}}$ See [link] .
• Approximate solutions of the equation $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}+d\text{\hspace{0.17em}}$ can be found using a graphing calculator. See [link] .
• The equation $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x},$ where $\text{\hspace{0.17em}}a>0,$ represents a vertical stretch if $\text{\hspace{0.17em}}|a|>1\text{\hspace{0.17em}}$ or compression if $\text{\hspace{0.17em}}0<|a|<1\text{\hspace{0.17em}}$ of the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}.\text{\hspace{0.17em}}$ See [link] .
• When the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ is multiplied by $\text{\hspace{0.17em}}-1,$ the result, $\text{\hspace{0.17em}}f\left(x\right)=-{b}^{x},$ is a reflection about the x -axis. When the input is multiplied by $\text{\hspace{0.17em}}-1,$ the result, $\text{\hspace{0.17em}}f\left(x\right)={b}^{-x},$ is a reflection about the y -axis. See [link] .
• All translations of the exponential function can be summarized by the general equation $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x+c}+d.\text{\hspace{0.17em}}$ See [link] .
• Using the general equation $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x+c}+d,$ we can write the equation of a function given its description. See [link] .

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
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Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
hi
salma
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opoku
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Ali
greetings from Iran
Ali
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Nharnhar