Graphs of exponential equations of the form $y=a\left(B^{x-h}\right)+k$y=a(Bx−h)+k (where $a$a, $B$B, and $h$h and $k$k are any number and $B>0$B>0) are called exponential graphs.

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

Transformations of exponential graphs

An exponential graph can be vertically translated by increasing or decreasing the $y$y-values by a constant number. So to translate $y=2^x$y=2x up by $k$k units gives us $y=2^x+k$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)

Similarly, an exponential graph can be horizontally translated by increasing or decreasing the $x$x-values by a constant number. However, the $x$x-value together with the translation must both be in the index. That is, to translate $y=2^x$y=2x to the left by $h$h units we get $y=2^{x+h}$y=2x+h.

Horizontally translating left by $2$2 ($y=2^{x+2}$y=2x+2) and right by $2$2 ($y=2^{x-2}$y=2x−2)

An exponential graph can be vertically scaled by multiplying every $y$y-value by a constant number. So to expand the exponential graph $y=2^x$y=2x by a scale factor of $a$a we get $y=a\left(2^x\right)$y=a(2x). We can compress an exponential graph by dividing by the scale factor instead.

Vertically expanding by a scale factor of $4$4 ($y=4\left(2^x\right)$y=4(2x)) and compressing by a scale factor of $2$2 ($y=\frac{1}{4}\left(2^x\right)$y=14(2x)). Notice that in this case, we get equivalent graphs from the horizontal translations $y=2^{x+2}$y=2x+2 and $2^{x-2}$2x−2 respectively. This is a consequence of the index laws.

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. Notice that this is distinct from $y=\left(-2\right)^x$y=(−2)x which will not give us an exponential graph.

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.

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

The graph of an exponential equation of the form $y=A\left(B^{x-h}\right)+k$y=A(Bx−h)+k is an exponential graph.

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 the following ways (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
Horizontally translated by $h$h units: $y=2^{x-h}$y=2x−h
Vertically scaled by a scale factor of $a$a: $y=a\left(2^x\right)$y=a(2x)
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.

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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.

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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.

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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.

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11.02 Exponential graphs (Enrichment)