# 4.2 Graphs of exponential functions

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• Graph exponential functions.
• Graph exponential functions using transformations.

As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

## Graphing exponential functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ whose base is greater than one. We’ll use the function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}.\text{\hspace{0.17em}}$ Observe how the output values in [link] change as the input increases by $\text{\hspace{0.17em}}1.$

 $x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $f\left(x\right)={2}^{x}$ $\frac{1}{8}$ $\frac{1}{4}$ $\frac{1}{2}$ $1$ $2$ $4$ $8$

Each output value is the product of the previous output and the base, $\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ We call the base $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ the constant ratio . In fact, for any exponential function with the form $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x},$ $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of $\text{\hspace{0.17em}}a.$

Notice from the table that

• the output values are positive for all values of $x;$
• as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases, the output values increase without bound; and
• as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases, the output values grow smaller, approaching zero.

[link] shows the exponential growth function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}.$ Notice that the graph gets close to the x -axis, but never touches it.

The domain of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ is all real numbers, the range is $\text{\hspace{0.17em}}\left(0,\infty \right),$ and the horizontal asymptote is $\text{\hspace{0.17em}}y=0.$

To get a sense of the behavior of exponential decay , we can create a table of values for a function of the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ whose base is between zero and one. We’ll use the function $\text{\hspace{0.17em}}g\left(x\right)={\left(\frac{1}{2}\right)}^{x}.\text{\hspace{0.17em}}$ Observe how the output values in [link] change as the input increases by $\text{\hspace{0.17em}}1.$

 $x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $g\left(x\right)=\left(\frac{1}{2}{\right)}^{x}$ $8$ $4$ $2$ $1$ $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{8}$

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio $\text{\hspace{0.17em}}\frac{1}{2}.$

Notice from the table that

• the output values are positive for all values of $\text{\hspace{0.17em}}x;$
• as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases, the output values grow smaller, approaching zero; and
• as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases, the output values grow without bound.

[link] shows the exponential decay function, $\text{\hspace{0.17em}}g\left(x\right)={\left(\frac{1}{2}\right)}^{x}.$

The domain of $\text{\hspace{0.17em}}g\left(x\right)={\left(\frac{1}{2}\right)}^{x}\text{\hspace{0.17em}}$ is all real numbers, the range is $\text{\hspace{0.17em}}\left(0,\infty \right),$ and the horizontal asymptote is $\text{\hspace{0.17em}}y=0.$

## Characteristics of the graph of the parent function f ( x ) = b x

An exponential function with the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x},$ $\text{\hspace{0.17em}}b>0,$ $\text{\hspace{0.17em}}b\ne 1,$ has these characteristics:

• one-to-one function
• horizontal asymptote: $\text{\hspace{0.17em}}y=0$
• domain:
• range: $\text{\hspace{0.17em}}\left(0,\infty \right)$
• x- intercept: none
• y- intercept: $\text{\hspace{0.17em}}\left(0,1\right)\text{\hspace{0.17em}}$
• increasing if $\text{\hspace{0.17em}}b>1$
• decreasing if $\text{\hspace{0.17em}}b<1$

[link] compares the graphs of exponential growth    and decay functions.

Given an exponential function of the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x},$ graph the function.

1. Create a table of points.
2. Plot at least $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ point from the table, including the y -intercept $\text{\hspace{0.17em}}\left(0,1\right).$
3. Draw a smooth curve through the points.
4. State the domain, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ the range, $\text{\hspace{0.17em}}\left(0,\infty \right),$ and the horizontal asymptote, $\text{\hspace{0.17em}}y=0.$

find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what 