To solve the problem of finding the equation of the new function when we vertically stretch the exponential function [tex]\( f(x) = 2^x \)[/tex] by a factor of 4, let's go through the process step-by-step:
1. Understand the original function: The original function is given as [tex]\( f(x) = 2^x \)[/tex]. This is an exponential function where the base is 2 and the exponent is [tex]\( x \)[/tex].
2. Apply the vertical stretch: A vertical stretch by a factor of 4 means that we need to multiply the entire function [tex]\( f(x) \)[/tex] by 4. This changes the y-values of the function but not the x-values.
In mathematical terms, if [tex]\( f(x) \)[/tex] is our original function, then the vertically stretched function [tex]\( g(x) \)[/tex] will be:
[tex]\[
g(x) = 4 \cdot f(x)
\][/tex]
3. Substitute the original function: We already know that [tex]\( f(x) = 2^x \)[/tex]. Now, substitute [tex]\( 2^x \)[/tex] into the equation for [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 4 \cdot 2^x
\][/tex]
4. Form the new equation: After substitution, the newly formed equation after the vertical stretch by a factor of 4 is:
[tex]\[
g(x) = 4 \cdot 2^x \quad \text{or} \quad g(x) = 4(2^x)
\][/tex]
5. Identify the correct option: Now, compare this new equation with the given options:
- Option A: [tex]\( f(x) = 6^x \)[/tex]
- Option B: [tex]\( f(x) = 8^x \)[/tex]
- Option C: [tex]\( f(x) = 2^{(4x)} \)[/tex]
- Option D: [tex]\( f(x) = 4(2^x) \)[/tex]
The correct option representing the new function after applying the vertical stretch by a factor of 4 is:
[tex]\[
\boxed{D}
\][/tex]
