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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![### Piecewise Function and Graph Sketching
In this lesson, we will learn how to sketch the graph of a piecewise function. A piecewise function is composed of multiple sub-functions, each applying to a certain interval of the main function's domain. Let's consider the given piecewise function \( f(x) \):
\[
f(x) =
\begin{cases}
-2x + 2 & \text{if } x < 2 \\
-3x + 3 & \text{if } x \geq 2
\end{cases}
\]
### Graph Explanation
Below is a coordinate plane where the piecewise function \( f(x) \) will be plotted. The graph is divided into two parts according to the conditions \( x < 2 \) and \( x \geq 2 \).
### Steps to Sketch the Graph:
1. **Plotting the first sub-function \( -2x + 2 \) for \( x < 2 \)**:
- This is a line with a slope of -2 and a y-intercept of 2.
- Calculate a few points for \( x < 2 \):
- When \( x = 0 \), \( y = -2(0) + 2 = 2 \).
- When \( x = -1 \), \( y = -2(-1) + 2 = 4 \).
- When \( x = 1 \), \( y = -2(1) + 2 = 0 \).
- Draw this portion of the line up to but not including \( x = 2 \), as indicated by the condition \( x < 2 \).
2. **Plotting the second sub-function \( -3x + 3 \) for \( x \geq 2 \)**:
- This is a line with a slope of -3 and a y-intercept of 3.
- Calculate a few points for \( x \geq 2 \):
- When \( x = 2 \), \( y = -3(2) + 3 = -3 \).
- When \( x = 3 \), \( y = -3(3) + 3 = -6 \).
- When \( x = 4 \), \( y = -3(4) + 3 = -9 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec5383c7-ce71-4639-9c19-f5515f12eee5%2F1150c9d3-1aec-45be-b0ec-877a93fa6afb%2Fsr9cphe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise Function and Graph Sketching
In this lesson, we will learn how to sketch the graph of a piecewise function. A piecewise function is composed of multiple sub-functions, each applying to a certain interval of the main function's domain. Let's consider the given piecewise function \( f(x) \):
\[
f(x) =
\begin{cases}
-2x + 2 & \text{if } x < 2 \\
-3x + 3 & \text{if } x \geq 2
\end{cases}
\]
### Graph Explanation
Below is a coordinate plane where the piecewise function \( f(x) \) will be plotted. The graph is divided into two parts according to the conditions \( x < 2 \) and \( x \geq 2 \).
### Steps to Sketch the Graph:
1. **Plotting the first sub-function \( -2x + 2 \) for \( x < 2 \)**:
- This is a line with a slope of -2 and a y-intercept of 2.
- Calculate a few points for \( x < 2 \):
- When \( x = 0 \), \( y = -2(0) + 2 = 2 \).
- When \( x = -1 \), \( y = -2(-1) + 2 = 4 \).
- When \( x = 1 \), \( y = -2(1) + 2 = 0 \).
- Draw this portion of the line up to but not including \( x = 2 \), as indicated by the condition \( x < 2 \).
2. **Plotting the second sub-function \( -3x + 3 \) for \( x \geq 2 \)**:
- This is a line with a slope of -3 and a y-intercept of 3.
- Calculate a few points for \( x \geq 2 \):
- When \( x = 2 \), \( y = -3(2) + 3 = -3 \).
- When \( x = 3 \), \( y = -3(3) + 3 = -6 \).
- When \( x = 4 \), \( y = -3(4) + 3 = -9 \
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