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.
