Transformations of Exponential Graphs (1 transform only)

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.

 

Example 1

adding a number

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.

 

Example 2

multiplying by a number

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.

Worked Examples

Question 1

Question 2

Question 3