9.02 Exponential graphs

Lesson

Worksheet

Practice

Lesson

Graphs of exponential equations such as $y=3^x$y=3x have the $x$x in the power. These types of functions are called exponential functions.

The exponential graph defined by $y=2^x$y=2x

Features of exponential graphs

Like lines, exponential graphs will always have a $y$y-intercept. This is the point on the graph which touches the $y$y-axis. We can find this by setting $x=0$x=0 and finding the value of $y$y. For example, the $y$y-intercept of $y=2^x$y=2x is $\left(0,1\right)$(0,1)

Similarly, we can look for $x$x-intercepts by setting $y=0$y=0 and then solving for $x$x. Because this is an exponential equation, there could be $0$0 or $1$1 solutions, and there will be the same number of $x$x-intercepts. For example, the graph of $y=2^x$y=2x has no $x$x-intercept.

Exponential graphs have a horizontal asymptote which is the horizontal line which the graph approaches but does not touch. For example, the horizontal asymptote of $y=2^x$y=2x is $y=0$y=0

Translate exponential graphs up and down the y-axis

An exponential graph can be vertically translated (moved up and down) by increasing or decreasing the $y$y-values by a constant number. So to move $y=2^x$y=2x up by $k$k units we use the function $y=2^x+k$y=2x+k.

Vertically translating up by $2$2 ($y=2^x+2$y=2x+2) and down by $2$2 ($y=2^x-2$y=2x−2

Reflect exponential graphs about the x or y-axis

We can vertically reflect an exponential graph about the $x$x-axis by taking the negative of the $y$y-values. So to reflect $y=2^x$y=2x about the $x$x-axis gives us $y=-2^x$y=−2x.

We can similarly horizontally reflect an exponential graph about the $y$y-axis by taking the negative of the $x$x-values. So to reflect $y=2^x$y=2x about the $y$y-axis gives us $y=2^{-x}$y=2−x. Note that this is the same function as $y=(\frac{1}{2})^x$y=(12)x . Why?

Reflecting the exponential graph about the $y$y-axis ($y=2^{-x}$y=2−x) and about the $x$x-axis ($y=-2^x$y=−2x)

Summary

Exponential graphs have a $y$y-intercept and can have $0$0 or $1$1$x$x-intercepts, depending on the solutions to the exponential equation.

Exponential graphs have a horizontal asymptote which is the horizontal line that the graph approaches but does not intersect.

Exponential graphs can be transformed in a number of ways including the following (starting with the exponential graph defined by $y=2^x$y=2x):

Vertically translated by $k$k units: $y=2^x+k$y=2x+k
Vertically reflected about the $x$x-axis: $y=-2^x$y=−2x
Horizontally reflected about the $y$y-axis: $y=2^{-x}$y=2−x

Practice questions

Question 1

Consider the equation $y=4^x$y=4x.

Complete the table of values.

$x$x $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3

$y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Using some of these points, graph the equation $y=4^x$y=4x on the number plane.

Loading Graph...
Which of the options completes the statement?

As $x$x increases, the $y$y-values
increase
A
decrease
B
stay the same
C
Which of the options completes the statement?

As $x$x decreases, the $y$y-values
increase
A
decrease
B
stay the same
C
Which of the following statements is true?
The curve crosses the $x$x-axis at a very small $x$x-value that is beyond the scale of the graph shown.
A
The curve never crosses the $x$x-axis.
B
The curve crosses the $x$x-axis at exactly one point on the graph shown.
C
At what value of $y$y does the graph cross the $y$y-axis?

$x$`x`	$-3$−3	$-2$−2	$-1$−1	$0$0	$1$1	$2$2	$3$3
$y$`y`	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Question 2

Consider the function $y=3^{-x}$y=3−x :

Find the $y$y-value of the $y$y-intercept of the curve $y=3^{-x}$y=3−x.
Fill in the table of values for $y=3^{-x}$y=3−x.

$x$x $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3

$y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Find the horizontal asymptote of the curve $y=3^{-x}$y=3−x.
Hence plot the curve $y=3^{-x}$y=3−x.

Loading Graph...
Is the function $y=3^{-x}$y=3−x, an increasing or decreasing function?
Increasing function
A
Decreasing function
B

Question 3

Consider the graph of $y=2^x$y=2x below.

Loading Graph...

How do we shift the graph of $y=2^x$y=2x to get the graph of $y=2^x-5$y=2x−5?
Move the graph upwards by $5$5 units.
A
Move the graph downwards by $5$5 units.
B
Move the graph $5$5 units to the left.
C
Move the graph $5$5 units to the right.
D
Hence plot $y=2^x-5$y=2x−5.
The graph of $y=2^x$y=2x is shown for reference.

Loading Graph...