Exponential Functions: Practice

Question 2 of 5

Select the correct answer.

If the graph of the function [tex]f(x) = e^x[/tex] is shifted 6 units to the right, what is the equation of the new graph?

A. [tex]g(x) = e^x + 6[/tex]
B. [tex]g(x) = e^{x + 6}[/tex]
C. [tex]g(x) = e^x - 6[/tex]
D. [tex]g(x) = e^{x - 6}[/tex]



Answer :

To find the equation of a function that has been shifted horizontally, we follow these steps:

1. Understand the Original Function: The original function given is [tex]\( f(x) = e^x \)[/tex].

2. Identify the Shift: The graph is shifted 6 units to the right.

3. Apply the Horizontal Shift: When a function [tex]\( f(x) \)[/tex] is shifted to the right by [tex]\( a \)[/tex] units, the new function [tex]\( g(x) \)[/tex] is given by [tex]\( g(x) = f(x - a) \)[/tex]. Here, "[tex]\( a \)[/tex]" is the number of units of the shift.

4. Substitute the Shift Value: In this case, [tex]\( a = 6 \)[/tex]. So, we need to shift [tex]\( f(x) = e^x \)[/tex] right by 6 units. Therefore, [tex]\( g(x) = f(x - 6) \)[/tex].

5. Formulate the New Function: Substitute [tex]\( x - 6 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = e^{(x - 6)} \][/tex]

Thus, the equation of the new graph is [tex]\( g(x) = e^{x - 6} \)[/tex].

Hence, the correct answer is:
[tex]\[ \boxed{g(x) = e^{x-6}} \][/tex]