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

 Page 6 / 6

## Verbal

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

An asymptote is a line that the graph of a function approaches, as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

## Algebraic

The graph of $\text{\hspace{0.17em}}f\left(x\right)={3}^{x}\text{\hspace{0.17em}}$ is reflected about the y -axis and stretched vertically by a factor of $\text{\hspace{0.17em}}4.\text{\hspace{0.17em}}$ What is the equation of the new function, $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ State its y -intercept, domain, and range.

$g\left(x\right)=4{\left(3\right)}^{-x};\text{\hspace{0.17em}}$ y -intercept: $\text{\hspace{0.17em}}\left(0,4\right);\text{\hspace{0.17em}}$ Domain: all real numbers; Range: all real numbers greater than $\text{\hspace{0.17em}}0.$

The graph of $\text{\hspace{0.17em}}f\left(x\right)={\left(\frac{1}{2}\right)}^{-x}\text{\hspace{0.17em}}$ is reflected about the y -axis and compressed vertically by a factor of $\text{\hspace{0.17em}}\frac{1}{5}.\text{\hspace{0.17em}}$ What is the equation of the new function, $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ State its y -intercept, domain, and range.

The graph of $\text{\hspace{0.17em}}f\left(x\right)={10}^{x}\text{\hspace{0.17em}}$ is reflected about the x -axis and shifted upward $\text{\hspace{0.17em}}7\text{\hspace{0.17em}}$ units. What is the equation of the new function, $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ State its y -intercept, domain, and range.

$g\left(x\right)=-{10}^{x}+7;\text{\hspace{0.17em}}$ y -intercept: $\text{\hspace{0.17em}}\left(0,6\right);\text{\hspace{0.17em}}$ Domain: all real numbers; Range: all real numbers less than $\text{\hspace{0.17em}}7.$

The graph of $\text{\hspace{0.17em}}f\left(x\right)={\left(1.68\right)}^{x}\text{\hspace{0.17em}}$ is shifted right $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ units, stretched vertically by a factor of $\text{\hspace{0.17em}}2,$ reflected about the x -axis, and then shifted downward $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ units. What is the equation of the new function, $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ State its y -intercept (to the nearest thousandth), domain, and range.

The graph of $\text{\hspace{0.17em}}f\left(x\right)=-\frac{1}{2}{\left(\frac{1}{4}\right)}^{x-2}+4\text{\hspace{0.17em}}$ is shifted left $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ units, stretched vertically by a factor of $\text{\hspace{0.17em}}4,$ reflected about the x -axis, and then shifted downward $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ units. What is the equation of the new function, $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ State its y -intercept, domain, and range.

$g\left(x\right)=2{\left(\frac{1}{4}\right)}^{x};\text{\hspace{0.17em}}$ y -intercept: Domain: all real numbers; Range: all real numbers greater than $\text{\hspace{0.17em}}0.$

## Graphical

For the following exercises, graph the function and its reflection about the y -axis on the same axes, and give the y -intercept.

$f\left(x\right)=3{\left(\frac{1}{2}\right)}^{x}$

$g\left(x\right)=-2{\left(0.25\right)}^{x}$

y -intercept: $\text{\hspace{0.17em}}\left(0,-2\right)$

$h\left(x\right)=6{\left(1.75\right)}^{-x}$

For the following exercises, graph each set of functions on the same axes.

$f\left(x\right)=3{\left(\frac{1}{4}\right)}^{x},$ $g\left(x\right)=3{\left(2\right)}^{x},$ and $\text{\hspace{0.17em}}h\left(x\right)=3{\left(4\right)}^{x}$

$f\left(x\right)=\frac{1}{4}{\left(3\right)}^{x},$ $g\left(x\right)=2{\left(3\right)}^{x},$ and $\text{\hspace{0.17em}}h\left(x\right)=4{\left(3\right)}^{x}$

For the following exercises, match each function with one of the graphs in [link] .

$f\left(x\right)=2{\left(0.69\right)}^{x}$

B

$f\left(x\right)=2{\left(1.28\right)}^{x}$

$f\left(x\right)=2{\left(0.81\right)}^{x}$

A

$f\left(x\right)=4{\left(1.28\right)}^{x}$

$f\left(x\right)=2{\left(1.59\right)}^{x}$

E

$f\left(x\right)=4{\left(0.69\right)}^{x}$

For the following exercises, use the graphs shown in [link] . All have the form $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x}.$

Which graph has the largest value for $\text{\hspace{0.17em}}b?$

D

Which graph has the smallest value for $\text{\hspace{0.17em}}b?$

Which graph has the largest value for $\text{\hspace{0.17em}}a?$

C

Which graph has the smallest value for $\text{\hspace{0.17em}}a?$

For the following exercises, graph the function and its reflection about the x -axis on the same axes.

$f\left(x\right)=\frac{1}{2}{\left(4\right)}^{x}$

$f\left(x\right)=3{\left(0.75\right)}^{x}-1$

$f\left(x\right)=-4{\left(2\right)}^{x}+2$

For the following exercises, graph the transformation of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}.\text{\hspace{0.17em}}$ Give the horizontal asymptote, the domain, and the range.

$f\left(x\right)={2}^{-x}$

$h\left(x\right)={2}^{x}+3$

Horizontal asymptote: $\text{\hspace{0.17em}}h\left(x\right)=3;$ Domain: all real numbers; Range: all real numbers strictly greater than $\text{\hspace{0.17em}}3.$

$f\left(x\right)={2}^{x-2}$

For the following exercises, describe the end behavior of the graphs of the functions.

$f\left(x\right)=-5{\left(4\right)}^{x}-1$

As $x\to \infty$ , $f\left(x\right)\to -\infty$ ;
As $x\to -\infty$ , $f\left(x\right)\to -1$

$f\left(x\right)=3{\left(\frac{1}{2}\right)}^{x}-2$

$f\left(x\right)=3{\left(4\right)}^{-x}+2$

As $x\to \infty$ , $f\left(x\right)\to 2$ ;
As $x\to -\infty$ , $f\left(x\right)\to \infty$

For the following exercises, start with the graph of $\text{\hspace{0.17em}}f\left(x\right)={4}^{x}.\text{\hspace{0.17em}}$ Then write a function that results from the given transformation.

Shift $f\left(x\right)$ 4 units upward

Shift $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ 3 units downward

$f\left(x\right)={4}^{x}-3$

Shift $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ 2 units left

Shift $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ 5 units right

$f\left(x\right)={4}^{x-5}$

Reflect $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ about the x -axis

Reflect $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ about the y -axis

$f\left(x\right)={4}^{-x}$

For the following exercises, each graph is a transformation of $\text{\hspace{0.17em}}y={2}^{x}.\text{\hspace{0.17em}}$ Write an equation describing the transformation.

$y=-{2}^{x}+3$

For the following exercises, find an exponential equation for the graph.

$y=-2{\left(3\right)}^{x}+7$

## Numeric

For the following exercises, evaluate the exponential functions for the indicated value of $\text{\hspace{0.17em}}x.$

$g\left(x\right)=\frac{1}{3}{\left(7\right)}^{x-2}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}g\left(6\right).$

$g\left(6\right)=800+\frac{1}{3}\approx 800.3333$

$f\left(x\right)=4{\left(2\right)}^{x-1}-2\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}f\left(5\right).$

$h\left(x\right)=-\frac{1}{2}{\left(\frac{1}{2}\right)}^{x}+6\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}h\left(-7\right).$

$h\left(-7\right)=-58$

## Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x}+d.$

$-50=-{\left(\frac{1}{2}\right)}^{-x}$

$116=\frac{1}{4}{\left(\frac{1}{8}\right)}^{x}$

$x\approx -2.953$

$12=2{\left(3\right)}^{x}+1$

$5=3{\left(\frac{1}{2}\right)}^{x-1}-2$

$x\approx -0.222$

$-30=-4{\left(2\right)}^{x+2}+2$

## Extensions

Explore and discuss the graphs of $\text{\hspace{0.17em}}F\left(x\right)={\left(b\right)}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}G\left(x\right)={\left(\frac{1}{b}\right)}^{x}.\text{\hspace{0.17em}}$ Then make a conjecture about the relationship between the graphs of the functions $\text{\hspace{0.17em}}{b}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\left(\frac{1}{b}\right)}^{x}\text{\hspace{0.17em}}$ for any real number $\text{\hspace{0.17em}}b>0.$

The graph of $\text{\hspace{0.17em}}G\left(x\right)={\left(\frac{1}{b}\right)}^{x}\text{\hspace{0.17em}}$ is the refelction about the y -axis of the graph of $\text{\hspace{0.17em}}F\left(x\right)={b}^{x};\text{\hspace{0.17em}}$ For any real number $\text{\hspace{0.17em}}b>0\text{\hspace{0.17em}}$ and function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x},$ the graph of $\text{\hspace{0.17em}}{\left(\frac{1}{b}\right)}^{x}\text{\hspace{0.17em}}$ is the the reflection about the y -axis, $\text{\hspace{0.17em}}F\left(-x\right).$

Prove the conjecture made in the previous exercise.

Explore and discuss the graphs of $\text{\hspace{0.17em}}f\left(x\right)={4}^{x},$ $\text{\hspace{0.17em}}g\left(x\right)={4}^{x-2},$ and $\text{\hspace{0.17em}}h\left(x\right)=\left(\frac{1}{16}\right){4}^{x}.\text{\hspace{0.17em}}$ Then make a conjecture about the relationship between the graphs of the functions $\text{\hspace{0.17em}}{b}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(\frac{1}{{b}^{n}}\right){b}^{x}\text{\hspace{0.17em}}$ for any real number n and real number $\text{\hspace{0.17em}}b>0.$

The graphs of $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ are the same and are a horizontal shift to the right of the graph of $\text{\hspace{0.17em}}f\left(x\right);\text{\hspace{0.17em}}$ For any real number n , real number $\text{\hspace{0.17em}}b>0,$ and function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x},$ the graph of $\text{\hspace{0.17em}}\left(\frac{1}{{b}^{n}}\right){b}^{x}\text{\hspace{0.17em}}$ is the horizontal shift $\text{\hspace{0.17em}}f\left(x-n\right).$

Prove the conjecture made in the previous exercise.

#### Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-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
Alfred Reply
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?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
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
Adu
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
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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
kinnecy Reply
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.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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