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.

The given function is f(x)=2x. a) Express the function h(x)=82x in terms of the original function…

Answered: Given the function f (x) = 2ª, along…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

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.

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