Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Question 8
**Describe the transformation to the graph of a toolkit function.**
Given function:
\[ h(x) = \left(\frac{1}{3} x \right)^3 - 3 \]
### Explanation:
The given function \( h(x) \) can be analyzed in terms of transformations applied to the basic cubic function \( f(x) = x^3 \). Here are the transformations described step-by-step:
1. **Horizontal Stretch:**
- The term \(\frac{1}{3} x\) inside the cubic function represents a horizontal stretch.
- A horizontal stretch by a factor of 3 means that every x-coordinate of the original function \( f(x) = x^3 \) is stretched away from the y-axis by a factor of 3.
2. **Vertical Shift:**
- The term -3 subtracted from \(\left(\frac{1}{3} x \right)^3\) indicates a vertical shift.
- Specifically, this represents a downward shift of the graph by 3 units.
### Summary of Transformations:
- **Horizontal Stretch by a factor of 3:**
The function \( \left(\frac{1}{3} x \right)^3 \) means that the input x is divided by 3, spreading out the graph horizontally.
- **Vertical Shift Downward by 3 units:**
The -3 outside the cubic term shifts the entire graph downward by 3 units.
### Transformed Function Graph:
Comparing \( h(x) \) with \( f(x) = x^3 \):
- The graph of \( f(x) = x^3 \) is horizontally stretched by a factor of 3.
- The graph is then moved down 3 units.
These transformations modify the basic shape and position of the cubic graph to form the graph of the given function \( h(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5fe3654-2655-4e19-bbf9-3baa2703a47c%2Fc260ed40-c56e-4725-9c59-3654f3b6ecc7%2Fgje1qy9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 8
**Describe the transformation to the graph of a toolkit function.**
Given function:
\[ h(x) = \left(\frac{1}{3} x \right)^3 - 3 \]
### Explanation:
The given function \( h(x) \) can be analyzed in terms of transformations applied to the basic cubic function \( f(x) = x^3 \). Here are the transformations described step-by-step:
1. **Horizontal Stretch:**
- The term \(\frac{1}{3} x\) inside the cubic function represents a horizontal stretch.
- A horizontal stretch by a factor of 3 means that every x-coordinate of the original function \( f(x) = x^3 \) is stretched away from the y-axis by a factor of 3.
2. **Vertical Shift:**
- The term -3 subtracted from \(\left(\frac{1}{3} x \right)^3\) indicates a vertical shift.
- Specifically, this represents a downward shift of the graph by 3 units.
### Summary of Transformations:
- **Horizontal Stretch by a factor of 3:**
The function \( \left(\frac{1}{3} x \right)^3 \) means that the input x is divided by 3, spreading out the graph horizontally.
- **Vertical Shift Downward by 3 units:**
The -3 outside the cubic term shifts the entire graph downward by 3 units.
### Transformed Function Graph:
Comparing \( h(x) \) with \( f(x) = x^3 \):
- The graph of \( f(x) = x^3 \) is horizontally stretched by a factor of 3.
- The graph is then moved down 3 units.
These transformations modify the basic shape and position of the cubic graph to form the graph of the given function \( h(x) \).
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