Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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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 \( x \geq 2 \).

![Graph](Image)

### 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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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 \( x \geq 2 \). ![Graph](Image) ### 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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