Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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]
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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.