Any function can be transformed by adding something to it and any function can be transformed by multiplying it by a number. We might write
$f(x)\rightarrow g(x)=f(x)+c$f(x)→g(x)=f(x)+c
$f(x)\rightarrow h(x)=af(x)$f(x)→h(x)=af(x)
We apply this general principle to exponential functions. Thus, if $f(x)=2^x$f(x)=2x then $g(x)$g(x) might be $2^x+5$2x+5 and $h(x)$h(x) might be $-\frac{1}{2}\cdot2^x$−12·2x.
The function $g(x)=2^x+5$g(x)=2x+5 is just the function $f(x)$f(x) with $5$5 added to every function value. The graph of $g(x)$g(x) must look the same as the graph of $f(x)$f(x) but shifted $5$5 units up the vertical axis. The following diagram shows these two functions.
Observe that $f(0)=1$f(0)=1 and $g(0)=1+5=6$g(0)=1+5=6, as expected.
The function $f(x)$f(x) is asymptotic to the horizontal axis and $g(x)$g(x) is asymptotic to the line $y(x)=5$y(x)=5.
At every point $x$x, the distance between $f(x)$f(x) and $g(x)$g(x) is $5$5.
The function $h(x)=-\frac{1}{2}\cdot2^x$h(x)=−12·2x, is the function $f(x)$f(x) with every function value multiplied by $-\frac{1}{2}$−12.
Multiplication by $\frac{1}{2}$12 brings all values of $f(x)$f(x) closer to zero by that factor. The graph of $h(x)$h(x) will appear compressed in the vertical direction compared with the graph of $f(x)$f(x).
Since all the values of $f(x)$f(x) are positive, all the values of $h(x)$h(x) must be negative. That is, the graph of $h(x)$h(x) is not only compressed in the vertical direction but is also reflected across the horizontal axis.
The graphs are represented in the following diagram.
Observe that $f(0)=1$f(0)=1 but $h(0)=-\frac{1}{2}$h(0)=−12; $f(1)=2$f(1)=2 but $h(1)=-1$h(1)=−1; $f(2)=4$f(2)=4 but $h(2)=-2$h(2)=−2; and so on, as expected.
Answer the following.
Determine the $y$y-intercept of $y=2^x$y=2x.
Hence or otherwise determine the $y$y-intercept of $y=2^x-2$y=2x−2.
Determine the horizontal asymptote of $y=2^x$y=2x.
Hence or otherwise determine the horizontal asymptote of $y=2^x-2$y=2x−2.
Consider a graph of $y=5^x$y=5x.
How could the graph of $y=-5^x$y=−5x be obtained from the graph of $y=5^x$y=5x?
through a vertical translation
through a reflection across the $y$y-axis
through a reflection across the $x$x-axis
by making it steeper
Given the graph of $y=5^x$y=5x, sketch $y=-5^x$y=−5x on the same coordinate plane.
This is a graph of $y=3^x$y=3x.
How do we shift the graph of $y=3^x$y=3x to get the graph of $y=3^x-4$y=3x−4?
Move the graph $4$4 units to the right.
Move the graph downwards by $4$4 units.
Move the graph $4$4 units to the left.
Move the graph upwards by $4$4 units.
Hence, plot $y=3^x-4$y=3x−4 on the same graph as $y=3^x$y=3x.