To determine the equation of the new graph [tex]\( g(x) \)[/tex] after applying a vertical stretch by a factor of 5 to the original function [tex]\( f(x) = 7^x \)[/tex], follow these steps:
1. Understanding the original function [tex]\( f(x) \)[/tex]:
The given function is [tex]\( f(x) = 7^x \)[/tex].
2. Applying a vertical stretch:
A vertical stretch by a factor of [tex]\( a \)[/tex] transforms the function [tex]\( f(x) \)[/tex] to [tex]\( a \cdot f(x) \)[/tex]. In this case, the stretch factor is 5.
3. Forming the new function [tex]\( g(x) \)[/tex]:
To apply a vertical stretch by a factor of 5 to [tex]\( f(x) = 7^x \)[/tex], multiply the entire function [tex]\( 7^x \)[/tex] by 5.
Therefore, the new function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = 5 \cdot 7^x
\][/tex]
Let's now identify the correct option:
A. [tex]\( g(x) = 5 \left( 7^x \right) \)[/tex] is correct.
B. [tex]\( g(x) = 5^{(7 x)} \)[/tex] is incorrect because this represents an exponential function with base 5 raised to the power of [tex]\( 7x \)[/tex].
C. [tex]\( g(x) = 7 \left( 5^x \right) \)[/tex] is incorrect because this represents the function with base 5 raised to the power of [tex]\( x \)[/tex], then multiplied by 7.
D. [tex]\( g(x) = 7^{(5 x)} \)[/tex] is incorrect because this represents an exponential function with base 7 raised to the power of [tex]\( 5x \)[/tex].
The correct answer is thus:
[tex]\[ \boxed{A} \][/tex]
