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Given the function f (x) = 2ª, along with its graph below, complete the following: (a) Express the new function h(x) = 8(2^) in terms of the original function f (X) = 2*. h(x) =

Given the function f (x) = 2ª, along with its graph below, complete the following: (a) Express the new function h(x) = 8(2^) in terms of the original function f (X) = 2*. h(x) =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Below is the transcription for the educational content: --- **Given the function \( f(x) = 2^x \), along with its graph below, complete the following:** **(a)** Express the new function \( h(x) = 8(2^x) \) in terms of the original function \( f(x) = 2^x \). \( h(x) = \) --- *Graph Explanation:* Even though the graph is not visible here, based on the problem, the graph of \( f(x) = 2^x \) would typically be an exponential curve starting at \( (0, 1) \) and rising sharply to the right. The graph of \( h(x) = 8(2^x) \) would be a vertical stretch of this function, scaling \( f(x) \) by a factor of 8, resulting in the function being 8 times taller at any given point \( x \).
Transcribed Image Text:Below is the transcription for the educational content: --- **Given the function \( f(x) = 2^x \), along with its graph below, complete the following:** **(a)** Express the new function \( h(x) = 8(2^x) \) in terms of the original function \( f(x) = 2^x \). \( h(x) = \) --- *Graph Explanation:* Even though the graph is not visible here, based on the problem, the graph of \( f(x) = 2^x \) would typically be an exponential curve starting at \( (0, 1) \) and rising sharply to the right. The graph of \( h(x) = 8(2^x) \) would be a vertical stretch of this function, scaling \( f(x) \) by a factor of 8, resulting in the function being 8 times taller at any given point \( x \).
(b) Find the \( y \)-intercept of the function \( h(x) = 8(2^x) \) and enter your answer as an ordered pair \( (x, y) \). Enter exact answers only, no approximations. \( y \)-intercept \( (x, y) = \) \( (0, 8) \) *You are correct. Your receipt no. is 157-7369.* (c) Transform the graph of \( f(x) = 2^x \) to get the graph of \( h(x) = 8(2^x) \). Use the \( y \)-intercept of \( h \) to verify that your transformation is correct. **Graph Description:** The graph shows two curves. The original function \( f(x) = 2^x \) is shown as a dashed red line that passes through the point (0, 1) on the y-axis, increasing exponentially as \( x \) increases. The transformed function \( h(x) = 8(2^x) \) is shown as a solid red curve passing through the point (0, 8) on the y-axis. This graph is a vertical stretch of the original function by a factor of 8. - There are annotations indicating the transformations: "Reflect through the y-axis" and "Reflect through the x-axis," which are not applicable here, as they are typically used for different kinds of transformations. - The graph features a coordinate grid with axes labeled and ticks marked, helping to illustrate the exponential growth of both functions.
Transcribed Image Text:(b) Find the \( y \)-intercept of the function \( h(x) = 8(2^x) \) and enter your answer as an ordered pair \( (x, y) \). Enter exact answers only, no approximations. \( y \)-intercept \( (x, y) = \) \( (0, 8) \) *You are correct. Your receipt no. is 157-7369.* (c) Transform the graph of \( f(x) = 2^x \) to get the graph of \( h(x) = 8(2^x) \). Use the \( y \)-intercept of \( h \) to verify that your transformation is correct. **Graph Description:** The graph shows two curves. The original function \( f(x) = 2^x \) is shown as a dashed red line that passes through the point (0, 1) on the y-axis, increasing exponentially as \( x \) increases. The transformed function \( h(x) = 8(2^x) \) is shown as a solid red curve passing through the point (0, 8) on the y-axis. This graph is a vertical stretch of the original function by a factor of 8. - There are annotations indicating the transformations: "Reflect through the y-axis" and "Reflect through the x-axis," which are not applicable here, as they are typically used for different kinds of transformations. - The graph features a coordinate grid with axes labeled and ticks marked, helping to illustrate the exponential growth of both functions.
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