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

$$f(x)→g(x)=f(x)+c
$$f(x)→h(x)=af(x)

We apply this general principle to exponential functions. Thus, if $$f(x)=2x then $$g(x) might be $$2x+5 and $$h(x) might be $$−12​·2x.

 

Example 1

adding a number

The function $$g(x)=2x+5 is just the function $$f(x) with $$5 added to every function value. The graph of $$g(x) must look the same as the graph of $$f(x) but shifted $$5 units up the vertical axis. The following diagram shows these two functions.

Observe that $$f(0)=1 and $$g(0)=1+5=6, as expected.

The function $$f(x) is asymptotic to the horizontal axis and $$g(x) is asymptotic to the line $$y(x)=5.

At every point $$x, the distance between $$f(x) and $$g(x) is $$5.

 

Example 2

multiplying by a number

The function  $$h(x)=−12​·2x, is the function $$f(x) with every function value multiplied by $$−12​.

Multiplication by $$12​ brings all values of $$f(x) closer to zero by that factor. The graph of $$h(x) will appear compressed in the vertical direction compared with the graph of $$f(x).

Since all the values of $$f(x) are positive, all the values of $$h(x) must be negative. That is, the graph of $$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 $$f(0)=1 but $$h(0)=−12​; $$f(1)=2 but $$h(1)=−1; $$f(2)=4 but $$h(2)=−2; and so on, as expected.

Worked Examples

Question 1

Question 2

Question 3