21.06 Exponential functions (Extended)
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
We can use this graph to approximate the solution to the equation $2^x=5$2x=5 by looking at the point of intersection where the curve crosses the line at $y=5,$y=5,which looks to be at about $(2.3,5)$(2.3,5). So the approximation of the solution to the equation is $x\approx2.3$x≈2.3.
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)
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.
Loading Graph...Which of the options completes the statement?
As $x$x increases, the $y$y-values
increase
Adecrease
Bstay the same
CWhich of the options completes the statement?
As $x$x decreases, the $y$y-values
increase
Adecrease
Bstay the same
CWhich 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.
AThe curve never crosses the $x$x-axis.
BThe curve crosses the $x$x-axis at exactly one point on the graph shown.
CAt what value of $y$y does the graph cross the $y$y-axis?
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
ADecreasing function
B
Question 3
Consider the graph of $y=2^x$y=2x below.
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.
AMove the graph downwards by $5$5 units.
BMove the graph $5$5 units to the left.
CMove the graph $5$5 units to the right.
DHence plot $y=2^x-5$y=2x−5.
The graph of $y=2^x$y=2x is shown for reference.Loading Graph...