**Graphing Polynomial Functions by Transformations:**To graph the function $ f(x) = \frac{1}{2}(x - 1)^5 - 2 $, we start with the basic graph of $ y = x^5 $ and apply transformations.**Steps:**1. **Horizontal Shift:** The graph of $ y = x^5 $ is shifted 1 unit to the right due to the $ (x - 1) $ inside the function.2. **Vertical Stretch/Compression:** There is a vertical compression by a factor of $ \frac{1}{2} $. This means the graph will be "squished" vertically, making it wider.3. **Vertical Shift:** Finally, the graph is shifted downward by 2 units due to the $- 2$ outside the function.**Graph Details:**- The coordinate system is a standard Cartesian plane with horizontal and vertical axes.- The graph is marked with grid lines to assist in accurately plotting the points and transformations.By following these transformations, you can accurately graph the function $ f(x) $.

Given: The function is f ( x ) = 12 x-15 - 2 . To sketch: Graph of f ( x ) = 12 x-15 - 2 using…

Answered: f(x) = (x – 1)5 – 2 -

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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**Graphing Polynomial Functions by Transformations:**

To graph the function $ f(x) = \frac{1}{2}(x - 1)^5 - 2 $, we start with the basic graph of $ y = x^5 $ and apply transformations.

**Steps:**

1. **Horizontal Shift:** The graph of $ y = x^5 $ is shifted 1 unit to the right due to the $ (x - 1) $ inside the function.

2. **Vertical Stretch/Compression:** There is a vertical compression by a factor of $ \frac{1}{2} $. This means the graph will be "squished" vertically, making it wider.

3. **Vertical Shift:** Finally, the graph is shifted downward by 2 units due to the $- 2$ outside the function.

**Graph Details:**

- The coordinate system is a standard Cartesian plane with horizontal and vertical axes.
- The graph is marked with grid lines to assist in accurately plotting the points and transformations.

By following these transformations, you can accurately graph the function $ f(x) $.

Expert Solution

Step 1

Given: The function is $f (x) = \frac{1}{2} {(x - 1)}^{5} - 2$ .

To sketch: Graph of $f (x) = \frac{1}{2} {(x - 1)}^{5} - 2$ using transformations in the graph of the function $y = x^{5}$ .

Concept used: Consider $f (x)$ be any function.

1. Horizontal translation: The curve of the function is moved $h$ units towards the right, thus the function becomes $f (x - h)$ .

2. Vertical translation: The curve of the function is moved $h$ units downwards, thus the function becomes $f (x) - h$ .

3. Horizontal dilation: If $0 < c < 1$ , then the curve of function is stretched by $c$ units, thus the function becomes $f (c x)$ and if $c > 1$ , then the curve of function is shrunk by $c$ units, and the function becomes $f (c x)$

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