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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, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, plus, 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:
The exponential function y=2^x+1 is graphed in the xy-plane. The graph is nonlinear and passes through the point (0, 2). The graph gains a steeper positive slope as x increases. As x decreases, the value of y approaches 1. As x increases, the value of y approaches infinity.
In this lesson, we'll learn to:
  1. Graph exponential functions
  2. 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, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, plus, 1, if we want to find the y-value when x, equals, 1, we can evaluate f, left parenthesis, 1, right parenthesis:
f(1)=21+1=3\begin{aligned} f(\purpleD{1}) &=2^\purpleD{1}+1 \\\\ &=3 \end{aligned}
Since f, left parenthesis, 1, right parenthesis, equals, 3, the point left parenthesis, 1, comma, 3, right parenthesis 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, equals, 0.
For example, the y-intercept of the graph of f, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, plus, 1 is:
f(0)=20+1=2\begin{aligned} f(\purpleD{0}) &=2^\purpleD{0}+1 \\\\ &=2 \end{aligned}

The slope

An exponential function is either always increasing or always decreasing. If you have already evaluated f, left parenthesis, 0, right parenthesis, try evaluating f, left parenthesis, 1, right parenthesis.
  • If f, left parenthesis, 1, right parenthesis, is greater than, f, left parenthesis, 0, right parenthesis, then the slope of the graph is positive.
  • If f, left parenthesis, 1, right parenthesis, is less than, f, left parenthesis, 0, right parenthesis, then the slope of the graph is negative.
For f, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, plus, 1, since f, left parenthesis, 0, right parenthesis, equals, 2 and f, left parenthesis, 1, right parenthesis, equals, 3, we can conclude that the slope of the graph is positive because 3, is greater than, 2.

The end behavior

End behavior is just another term for what happens to the value of y as 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, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, plus, 1:
  • As x increases, f, left parenthesis, x, right parenthesis becomes very large. The value of y on the right end of the graph approaches infinity.
  • As x decreases, f, left parenthesis, x, right parenthesis 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, left parenthesis, minus, 2, comma, 1, point, 25, right parenthesis, left parenthesis, minus, 1, comma, 1, point, 5, right parenthesis, and left parenthesis, 2, comma, 5, right parenthesis, we can sketch the graph of f, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, plus, 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.
The exponential function y=2^x+1 is graphed in the xy-plane. The graph contains the points (-2, 1.25), (-1, 1.5), (0, 2), (1, 3), and (2, 5) and has a positive slope. As x decreases, the value of y approaches the horizontal line y=1. As x increases, the value of y approaches infinity.
To graph an exponential function:
  1. Evaluate the function at various values of x—start with minus, 1, 0, and 1. Find additional points on the graph if necessary.
  2. Use the points from Step 1 to sketch a curve, establishing the y-intercept and the direction of the slope.
  3. 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, left parenthesis, x, right parenthesis, equals, 12, dot, left parenthesis, start fraction, 1, divided by, 2, end fraction, right parenthesis, start superscript, x, end superscript.

Try it!

try: find points on an exponential graph
f, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, minus, 1
Identify points on the graph of the exponential function above and completing the table below.
xf, left parenthesis, x, right parenthesis
minus, 3minus, 0, point, 875
minus, 1
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
0
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
11
2
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


try: describe an exponential graph
f, left parenthesis, x, right parenthesis, equals, 2, start superscript, x, end superscript, minus, 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
.
As 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, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript
Let's consider the case where b is a positive real number:
  • If b, is greater than, 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, is less than, b, is less than, 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 start color #7854ab, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, end color #7854ab and start color #ca337c, f, left parenthesis, x, right parenthesis, equals, left parenthesis, start fraction, 2, divided by, 3, end fraction, right parenthesis, start superscript, x, end superscript, end color #ca337c are shown below.
The graphs of y=1.5^x and y=(2/3)^x are shown in the xy-plane. For the graph of y=1.5^x, the value of y approaches infinity as x increases and approaches 0 as x decreases. For the graph of y=(2/3)^x, the value of y approaches 0 as x increases and approaches infinity as x decreases.

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, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, the value of y approaches infinity on one end and the constant 0 on the other.
  • For f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the value of y approaches infinity on one end and d on the other.
The graphs of start color #7854ab, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, end color #7854ab and start color #ca337c, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #ca337c are shown below.
  • For start color #7854ab, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, end color #7854ab, as x decreases, the value of y approaches start color #7854ab, 0, end color #7854ab.
  • For start color #ca337c, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #ca337c, as x decreases, the value of y approaches start color #ca337c, 2, end color #ca337c.
The graphs of y=1.5^x and y=1.5^x+2 are shown in the xy-plane. or the graph of y=1.5^x, the value of y approaches infinity as x increases and approaches 0 as x decreases. or the graph of y=1.5^x+2, the value of y approaches infinity as x increases and approaches 2 as x decreases.

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, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the y-intercept is 1, plus, d.
  • For f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, the y-intercept is a, dot, 1, equals, a. In this form, a is also called the initial value.
  • For f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, plus, d, the y-intercept is a, plus, d.
The graphs of start color #ca337c, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #ca337c, start color #208170, f, left parenthesis, x, right parenthesis, equals, 2, dot, 1, point, 5, start superscript, x, end superscript, end color #208170, and start color #a75a05, f, left parenthesis, x, right parenthesis, equals, 2, dot, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #a75a05 are shown below.
  • For start color #ca337c, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #ca337c, the y-intercept is start color #ca337c, 1, plus, 2, equals, 3, end color #ca337c.
  • For start color #208170, f, left parenthesis, x, right parenthesis, equals, 2, dot, 1, point, 5, start superscript, x, end superscript, end color #208170, the y-intercept is start color #208170, 2, dot, 1, equals, 2, end color #208170.
  • For start color #a75a05, f, left parenthesis, x, right parenthesis, equals, 2, dot, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #a75a05, the y-intercept is start color #a75a05, 2, plus, 2, equals, 4, end color #a75a05.
Three exponential functions are graphed in the xy-plane. The graph of y=1.5^x+2 has a y-intercept of 3. The graph of y=21.5^x has a y-intercept of 2. The graph of y=21.5^x+2 has a y-intercept of 4.

Try it!

TRY: identify the features of an exponential graph without finding points
f, left parenthesis, x, right parenthesis, equals, 64, left parenthesis, 0, point, 25, right parenthesis, start superscript, x, end superscript
Consider the exponential function f above. The y-intercept of its graph, or the initial value of the function, is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.
Because the base of the exponent, 0, point, 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, equals, 4, dot, 0, point, 5, start superscript, x, end superscript ?
Choose 1 answer:
Choose 1 answer:


Practice: transform an exponential function
An exponential curve in the xy-plane. The curve passes through the points (0, 1), (1, 2), and (2, 4). As x becomes more negative, y becomes closer to 0. As x becomes more positive, y becomes larger at an increasing rate.
The graph of function f is shown in the x, y-plane above. Which of the following is the graph of f, left parenthesis, x, right parenthesis, plus, 2 ?
Choose 1 answer:
Choose 1 answer:


Things to remember

For f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, where b is a positive real number:
  • If b, is greater than, 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, is less than, b, is less than, 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, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, the value of y approaches infinity on one end and the constant 0 on the other.
  • For f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the value of y approaches infinity on one end and d on the other.
To shift the y-intercept:
  • For f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the y-intercept is 1, plus, d.
  • For f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, the y-intercept is a, dot, 1, equals, a. In this form, a is also called the initial value.
  • For f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, plus, d, the y-intercept is a, plus, d.

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