Graphing exponential functions | Lesson

What are exponential functions, and how frequently do they appear on the test?

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

Graphing exponential growth & decay

Using points to sketch an exponential graph

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

The $y$y-intercept

The slope

• If $f(1)>f(0)$f, left parenthesis, 1, right parenthesis, is greater than, f, left parenthesis, 0, right parenthesis, then the slope of the graph is positive.
• If $f(1)<f(0)$f, left parenthesis, 1, right parenthesis, is less than, f, left parenthesis, 0, right parenthesis, then the slope of the graph is negative.

The end behavior

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

1. Evaluate the function at various values of $x$x—start with $-1$minus, 1, $0$0, and $1$1. Find additional points on the graph if necessary.
2. Use the points from Step 1 to sketch a curve, establishing the $y$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 $y$y-axis.

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

• If $b>1$b, is greater than, 1, then the slope of the graph is positive, and the graph shows exponential growth. As $x$x increases, the value of $y$y approaches infinity. As $x$x decreases, the value of $y$y approaches $0$0.
• If $0<b<1$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 $x$x increases, the value of $y$y approaches $0$0. As $x$x decreases, the value of $y$y approaches infinity.
• For all values of $b$b, the $y$y-intercept is $1$1.

How do we shift the horizontal asymptote?

• For $f(x)=b^x$f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, the value of $y$y approaches infinity on one end and the constant $0$0 on the other.
• For $f(x)=b^x+d$f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the value of $y$y approaches infinity on one end and $d$d on the other.

• For $\purpleD{f(x)=1.5^x}$start color #7854ab, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, end color #7854ab, as $x$x decreases, the value of $y$y approaches $\purpleD{0}$start color #7854ab, 0, end color #7854ab.
• For $\maroonD{f(x)=1.5^x+2}$start color #ca337c, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #ca337c, as $x$x decreases, the value of $y$y approaches $\maroonD{2}$start color #ca337c, 2, end color #ca337c.

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

• For $f(x)=b^x+d$f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the $y$y-intercept is $1+d$1, plus, d.
• For $f(x)=a\cdot b^x$f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, the $y$y-intercept is $a\cdot1=a$a, dot, 1, equals, a. In this form, $a$a is also called the initial value.
• For $f(x)=a\cdot b^x+d$f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, plus, d, the $y$y-intercept is $a+d$a, plus, d.

• For $\maroonD{f(x)=1.5^x+2}$start color #ca337c, f, left parenthesis, x, right parenthesis, equals, 1, point, 5, start superscript, x, end superscript, plus, 2, end color #ca337c, the $y$y-intercept is $\maroonD{1+2=3}$start color #ca337c, 1, plus, 2, equals, 3, end color #ca337c.
• For $\tealE{f(x)=2\cdot1.5^x}$start color #208170, f, left parenthesis, x, right parenthesis, equals, 2, dot, 1, point, 5, start superscript, x, end superscript, end color #208170, the $y$y-intercept is $\tealE{2\cdot1=2}$start color #208170, 2, dot, 1, equals, 2, end color #208170.
• For $\goldE{f(x)=2\cdot1.5^x+2}$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 $y$y-intercept is $\goldE{2+2=4}$start color #a75a05, 2, plus, 2, equals, 4, end color #a75a05.

Try it!

You turn!

Things to remember

• If $b>1$b, is greater than, 1, then the slope of the graph is positive, and the graph shows exponential growth. As $x$x increases, the value of $y$y approaches infinity. As $x$x decreases, the value of $y$y approaches $0$0.
• If $0<b<1$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 $x$x increases, the value of $y$y approaches $0$0. As $x$x decreases, the value of $y$y approaches infinity.
• For all values of $b$b, the $y$y-intercept is $1$1.

• For $f(x)=b^x$f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, the value of $y$y approaches infinity on one end and the constant $0$0 on the other.
• For $f(x)=b^x+d$f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the value of $y$y approaches infinity on one end and $d$d on the other.

• For $f(x)=b^x+d$f, left parenthesis, x, right parenthesis, equals, b, start superscript, x, end superscript, plus, d, the $y$y-intercept is $1+d$1, plus, d.
• For $f(x)=a\cdot b^x$f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, the $y$y-intercept is $a\cdot1=a$a, dot, 1, equals, a. In this form, $a$a is also called the initial value.
• For $f(x)=a\cdot b^x+d$f, left parenthesis, x, right parenthesis, equals, a, dot, b, start superscript, x, end superscript, plus, d, the $y$y-intercept is $a+d$a, plus, d.