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
Graph the function
![**Graph the Function**
We have a piecewise function \( f(x) \) defined as follows:
\[
f(x) =
\begin{cases}
x^2 - 4 & \text{for } x \leq 2 \\
3x - 6 & \text{for } x > 2
\end{cases}
\]
**Graph Description:**
The graph consists of two distinct parts based on the value of \( x \):
1. **For \( x \leq 2 \):** The function is defined by the quadratic equation \( x^2 - 4 \). This represents a parabola opening upwards, with a vertex at the point where \( x = 0 \), resulting in \( f(x) = -4 \).
2. **For \( x > 2 \):** The function is defined by the linear equation \( 3x - 6 \). This forms a straight line with a slope of 3, starting from just beyond \( x = 2 \).
**Interpretation of the Graph:**
- The left section of the graph, where \( x \leq 2 \), is a segment of a parabola.
- The right section of the graph, where \( x > 2 \), transitions to a straight line.
- The graph has a different slope when transitioning from the parabola to the line at \( x = 2 \).
- The graph is continuous but the derivative is not continuous at \( x = 2 \), which indicates a sharp change in the slope at this point.
This piecewise function effectively combines two different functional forms, utilizing the conditions \( x \leq 2 \) and \( x > 2 \) to determine which part of the function to apply.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a54d7d8-382f-4dae-bfd8-b87945b99138%2Fa8a7d409-a44d-4cf6-8fa5-f3549a698ebe%2Fr8669zm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graph the Function**
We have a piecewise function \( f(x) \) defined as follows:
\[
f(x) =
\begin{cases}
x^2 - 4 & \text{for } x \leq 2 \\
3x - 6 & \text{for } x > 2
\end{cases}
\]
**Graph Description:**
The graph consists of two distinct parts based on the value of \( x \):
1. **For \( x \leq 2 \):** The function is defined by the quadratic equation \( x^2 - 4 \). This represents a parabola opening upwards, with a vertex at the point where \( x = 0 \), resulting in \( f(x) = -4 \).
2. **For \( x > 2 \):** The function is defined by the linear equation \( 3x - 6 \). This forms a straight line with a slope of 3, starting from just beyond \( x = 2 \).
**Interpretation of the Graph:**
- The left section of the graph, where \( x \leq 2 \), is a segment of a parabola.
- The right section of the graph, where \( x > 2 \), transitions to a straight line.
- The graph has a different slope when transitioning from the parabola to the line at \( x = 2 \).
- The graph is continuous but the derivative is not continuous at \( x = 2 \), which indicates a sharp change in the slope at this point.
This piecewise function effectively combines two different functional forms, utilizing the conditions \( x \leq 2 \) and \( x > 2 \) to determine which part of the function to apply.
Expert Solution
Step 1
To graph the given function, first, we find out some points and then plot the points in the coordinate plane.
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