If you vertically stretch the exponential function [tex]f(x)=2^x[/tex] by a factor of 4, what is the equation of the new function?

A. [tex]f(x)=6^x[/tex]
B. [tex]f(x)=8^x[/tex]
C. [tex]f(x)=2^{(4 x)}[/tex]
D. [tex]f(x)=4\left(2^x\right)[/tex]



Answer :

To find the new function when the function [tex]\( f(x) = 2^x \)[/tex] is stretched vertically by a factor of 4, follow these steps:

1. Understand Vertical Stretch:
- Vertically stretching a function by a factor means multiplying the entire function by that factor. Here, the factor is 4.

2. Apply the Vertical Stretch:
- Start with the original function:
[tex]\[ f(x) = 2^x \][/tex]
- Multiply the entire function by 4 to apply the vertical stretch:
[tex]\[ g(x) = 4 \cdot 2^x \][/tex]

3. Express the New Function:
- The equation of the new function after the vertical stretch is:
[tex]\[ g(x) = 4 \left(2^x\right) \][/tex]

4. Match the Options:
- Compare the new function [tex]\( g(x) = 4 \left(2^x\right) \)[/tex] against the given options:
- A. [tex]\( f(x) = 6^x \)[/tex]
- B. [tex]\( f(x) = 8^x \)[/tex]
- C. [tex]\( f(x) = 2^{(4x)} \)[/tex]
- D. [tex]\( f(x) = 4 \left(2^x\right) \)[/tex]

5. Determine the Correct Option:
- The correct option is D, since it matches the new function exactly:
[tex]\[ f(x) = 4 \left(2^x\right) \][/tex]

Thus, the equation of the new function after vertically stretching [tex]\( f(x) = 2^x \)[/tex] by a factor of 4 is:

[tex]\[ \boxed{4 \left(2^x\right)} \][/tex]

And the correct answer is option D.