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Course: Digital SAT Math?>?Unit 4

Lesson 12: Exponential graphs: foundations

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Exponential graphs | Lesson

A guide to exponential graphs on the digital SAT

What are exponential graphs?

In an exponential function, the output of the function is based on an expression in which the input is in the exponent. For example,

f (x) = 2^{x} + 1

is an exponential function, because

x

is an exponent of the base

2

The graphs of exponential functions are nonlinear—because their slopes are always changing, they look like curves, not straight lines:

In this lesson, we'll learn to:

Graph exponential functions
Identify the features of exponential functions

You can learn anything. Let's do this!

How do I graph exponential functions, and what are their features?

Graphing exponential growth & decay

Khan Academy video wrapper

Graphing exponential growth & decaySee video transcript

Using points to sketch an exponential graph

The best way to graph exponential functions is to find a few points on the graph and to sketch the graph based on these points.

To find a point on the graph, select an input value and calculate the output value. For example, for the function

f (x) = 2^{x} + 1

, if we want to find the

y

-value when

x = 1

, we can evaluate

f (1)

\begin{aligned} f (1) & = 2^{1} + 1 \\ = 3 \end{aligned}

Since

f (1) = 3

, the point

(1, 3)

is a point on the graph.

We need to use the points to help us identify three important features of the graph:

What is the $y$ ?-intercept?
Is the slope of the graph positive or negative?
What happens to the value of $y$ ? as the value of $x$ ? becomes very large?

The $y$ ?-intercept

Not only is the

y

-intercept the easiest feature to identify, it also helps you figure out the rest of the features.

To find the

y

-intercept, evaluate the function at

x = 0

For example, the

y

-intercept of the graph of

f (x) = 2^{x} + 1

is:

\begin{aligned} f (0) & = 2^{0} + 1 \\ = 2 \end{aligned}

The slope

An exponential function is either always increasing or always decreasing. If you have already evaluated

f (0)

, try evaluating

f (1)

If $f (1) > f (0)$ ?, then the slope of the graph is positive.
If $f (1) < f (0)$ ?, then the slope of the graph is negative.

For

f (x) = 2^{x} + 1

, since

f (0) = 2

and

f (1) = 3

, we can conclude that the slope of the graph is positive because

3 > 2

The end behavior

End behavior is just another term for what happens to the value of

y

x

becomes very large in both the positive and negative directions. For the graph of an exponential function, the value of

y

will always grow to positive or negative infinity on one end and approach, but not reach, a horizontal line on the other. The horizontal line that the graph approaches but never reaches is called the horizontal asymptote.

For

f (x) = 2^{x} + 1

As $x$ ? increases, $f (x)$ ? becomes very large. The value of $y$ ? on the right end of the graph approaches infinity.
As $x$ ? decreases, $f (x)$ ? becomes closer and closer to $1$ ?, but it's always slightly larger than $1$ ?. The value of $y$ ? on the left end of the graph approaches, but never reaches, $1$ ?.

Putting it all together

With the help of a few more points,

(? 2, 1.25)

(? 1, 1.5)

, and

(2, 5)

, we can sketch the graph of

f (x) = 2^{x} + 1

Note: if you're graphing by hand, it's more important to recognize that the value of

y

will grow to positive infinity as

x

increases than getting the graph exactly right! You can use the points you identified to establish a trend and sketch out the curve.

To graph an exponential function:

Evaluate the function at various values of $x$ ?—start with $? 1$ ?, $0$ ?, and $1$ ?. Find additional points on the graph if necessary.
Use the points from Step 1 to sketch a curve, establishing the $y$ ?-intercept and the direction of the slope.
Extend the curve on both ends. One end will approach a horizontal asymptote, and the other will approach positive or negative infinity along the $y$ ?-axis.

Example: Graph

f (x) = 12 ? {(\frac{1}{2})}^{x}

First, let's find a few points. The table below shows the coordinates of several points on the graph.

$x$ ?	$f (x)$ ?
$? 1$ ?	$24$ ?
$0$ ?	$12$ ?
$1$ ?	$6$ ?
$2$ ?	$3$ ?
$5$ ?	$\frac{3}{8}$ ?

Next, let's graph these points and sketch a curve through them.

We can see that the

y

-intercept is located at

(0, 12)

, and the graph has a negative slope.

x

decreases,

y

appears to increase more rapidly toward infinity. As

x

increases,

y

appears to decrease toward

0

With these features in mind, we can confidently graph

f (x) = 12 ? {(\frac{1}{2})}^{x}

Try it!

try: find points on an exponential graph

f (x) = 2^{x} ? 1

Identify points on the graph of the exponential function above and completing the table below.

$x$ ?	$f (x)$ ?
$? 3$ ?	$? 0.875$ ?
$? 1$ ?	Your answer should be an integer, like $6$ ? a simplified proper fraction, like $3 / 5$ ? a simplified improper fraction, like $7 / 4$ ? a mixed number, like $1 3 / 4$ ? an exact decimal, like $0.75$ ? a multiple of pi, like $12 pi$ ? or $2 / 3 pi$ ?
$0$ ?	Your answer should be an integer, like $6$ ? a simplified proper fraction, like $3 / 5$ ? a simplified improper fraction, like $7 / 4$ ? a mixed number, like $1 3 / 4$ ? an exact decimal, like $0.75$ ? a multiple of pi, like $12 pi$ ? or $2 / 3 pi$ ?
$1$ ?	$1$ ?
$2$ ?	Your answer should be an integer, like $6$ ? a simplified proper fraction, like $3 / 5$ ? a simplified improper fraction, like $7 / 4$ ? a mixed number, like $1 3 / 4$ ? an exact decimal, like $0.75$ ? a multiple of pi, like $12 pi$ ? or $2 / 3 pi$ ?

try: describe an exponential graph

f (x) = 2^{x} ? 1

Using the points from the previous question, complete the following statements about the graph of the exponential function above.

The

y

-intercept of the graph is located at

x

increases,

y

As the value of

x

decreases, the value of

y

approaches, but never reaches,

How do I identify features of exponential graphs from exponential functions?

Graphs of exponential growth

Khan Academy video wrapper

Graphs of exponential growthSee video transcript

Identifying features of graphs from functions

The basic exponential function

Let's start with the basics!

The most basic exponential function has a base and an exponent:

f (x) = b^{x}

Let's consider the case where

b

is a positive real number:

If $b > 1$ ?, then the slope of the graph is positive, and the graph shows exponential growth. As $x$ ? increases, the value of $y$ ? approaches infinity. As $x$ ? decreases, the value of $y$ ? approaches $0$ ?.
If $0 < b < 1$ ?, then the slope of the graph is negative, and the graph shows exponential decay. In this case, as $x$ ? increases, the value of $y$ ? approaches $0$ ?. As $x$ ? decreases, the value of $y$ ? approaches infinity.
For all values of $b$ ?, the $y$ ?-intercept is $1$ ?.

The graphs of

f (x) = {1.5}^{x}

and

f (x) = {(\frac{2}{3})}^{x}

are shown below.

How do we shift the horizontal asymptote?

The

y

-value of every exponential graph approaches positive or negative infinity on one end and a constant on the other. We can change the constant value

y

approaches by introducing a constant term to the function:

For $f (x) = b^{x}$ ?, the value of $y$ ? approaches infinity on one end and the constant $0$ ? on the other.
For $f (x) = b^{x} + d$ ?, the value of $y$ ? approaches infinity on one end and $d$ ? on the other.

The graphs of

f (x) = {1.5}^{x}

and

f (x) = {1.5}^{x} + 2

are shown below.

For $f (x) = {1.5}^{x}$ ?, as $x$ ? decreases, the value of $y$ ? approaches $0$ ?.
For $f (x) = {1.5}^{x} + 2$ ?, as $x$ ? decreases, the value of $y$ ? approaches $2$ ?.

How do we shift the $y$ ?-intercept?

We can change the

y

-intercept of the graph either by introducing a constant term (as above) or introducing a coefficient for the exponential term:

For $f (x) = b^{x} + d$ ?, the $y$ ?-intercept is $1 + d$ ?.
For $f (x) = a ? b^{x}$ ?, the $y$ ?-intercept is $a ? 1 = a$ ?. In this form, $a$ ? is also called the initial value.
For $f (x) = a ? b^{x} + d$ ?, the $y$ ?-intercept is $a + d$ ?.

The graphs of

f (x) = {1.5}^{x} + 2

f (x) = 2 ? {1.5}^{x}

, and

f (x) = 2 ? {1.5}^{x} + 2

are shown below.

For $f (x) = {1.5}^{x} + 2$ ?, the $y$ ?-intercept is $1 + 2 = 3$ ?.
For $f (x) = 2 ? {1.5}^{x}$ ?, the $y$ ?-intercept is $2 ? 1 = 2$ ?.
For $f (x) = 2 ? {1.5}^{x} + 2$ ?, the $y$ ?-intercept is $2 + 2 = 4$ ?.

Try it!

TRY: identify the features of an exponential graph without finding points

f (x) = 64 (0.25)^{x}

Consider the exponential function

f

above. The

y

-intercept of its graph, or the initial value of the function, is

Because the base of the exponent,

0.25

, is less than

1

, the slope of the graph is

. As the value of

x

increases by

1

, the value of

y

You turn!

Practice: match an exponential function to its graph

Which of the following is the graph of the function

y = 4 ? {0.5}^{x}

Practice: transform an exponential function

The graph of function

f

is shown in the

x y

-plane above. Which of the following is the graph of

f (x) + 2

Things to remember

For

f (x) = b^{x}

, where

b

is a positive real number:

If $b > 1$ ?, then the slope of the graph is positive, and the graph shows exponential growth. As $x$ ? increases, the value of $y$ ? approaches infinity. As $x$ ? decreases, the value of $y$ ? approaches $0$ ?.
If $0 < b < 1$ ?, then the slope of the graph is negative, and the graph shows exponential decay. In this case, as $x$ ? increases, the value of $y$ ? approaches $0$ ?. As $x$ ? decreases, the value of $y$ ? approaches infinity.
For all values of $b$ ?, the $y$ ?-intercept is $1$ ?.

To shift the horizontal asymptote:

For $f (x) = b^{x}$ ?, the value of $y$ ? approaches infinity on one end and the constant $0$ ? on the other.
For $f (x) = b^{x} + d$ ?, the value of $y$ ? approaches infinity on one end and $d$ ? on the other.

To shift the

y

-intercept:

For $f (x) = b^{x} + d$ ?, the $y$ ?-intercept is $1 + d$ ?.
For $f (x) = a ? b^{x}$ ?, the $y$ ?-intercept is $a ? 1 = a$ ?. In this form, $a$ ? is also called the initial value.
For $f (x) = a ? b^{x} + d$ ?, the $y$ ?-intercept is $a + d$ ?.

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Digital SAT Math

Course: Digital SAT Math?>?Unit 4

What are exponential graphs?

How do I graph exponential functions, and what are their features?

Graphing exponential growth & decay

Using points to sketch an exponential graph

The y? -intercept

The slope

The end behavior

Putting it all together

Try it!

How do I identify features of exponential graphs from exponential functions?

Graphs of exponential growth

Identifying features of graphs from functions

The basic exponential function

How do we shift the horizontal asymptote?

How do we shift the y? -intercept?

Try it!

You turn!

Things to remember

Want to join the conversation?

The $y$ ?-intercept

How do we shift the $y$ ?-intercept?