Sketch the graph of the function. (3x + 3, X ≤ -1 k(x) = 3x²-2, -1 1 y
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Piecewise Function and Its Graph
#### Function Definition:
\[ k(x) =
\begin{cases}
3x + 3 & \text{if } x \leq -1 \\
3x^2 - 2 & \text{if } -1 < x \leq 1 \\
3 - x^2 & \text{if } x > 1
\end{cases}
\]
#### Explanation of the Graph:
The graph of the piecewise function \( k(x) \) is divided into three distinct parts, corresponding to different intervals for \( x \):
1. **For \( x \leq -1 \):**
- The function \( k(x) \) is given by \( 3x + 3 \).
- This part of the graph is a straight line with a slope of 3 and a y-intercept at 3. On the graph, this is represented by a line segment extending to the left from the point \((-1, 0)\).
2. **For \( -1 < x \leq 1 \):**
- The function \( k(x) \) is given by \( 3x^2 - 2 \).
- This part of the graph is a parabola opening upwards. It intersects the y-axis at \((0, -2)\). Moving from left to right, it starts at \((-1, 1)\), reaches the vertex at \((0, -2)\), and continues to \((1, 1)\).
3. **For \( x > 1 \):**
- The function \( k(x) \) is given by \( 3 - x^2 \).
- This part of the graph is a downward-opening parabola. Starting from the point \((1, 2)\), it extends to the right, decreasing as \( x \) increases.
#### Key Points on the Graph:
- There is an open circle at the point \((-1, 0)\) indicating that this point is not included in the \( 3x + 3 \) segment.
- Another open circle is at the point \((1, 2)\) where the parabola \( 3 - x^2 \) starts but does not include this point.
By carefully analyzing these segments, one can understand the behavior of the piecewise function](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86020f22-4efe-437b-8b05-308770976ff7%2F57a8960a-ca37-433b-916b-a6333aafda2c%2F9i8xz4r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise Function and Its Graph
#### Function Definition:
\[ k(x) =
\begin{cases}
3x + 3 & \text{if } x \leq -1 \\
3x^2 - 2 & \text{if } -1 < x \leq 1 \\
3 - x^2 & \text{if } x > 1
\end{cases}
\]
#### Explanation of the Graph:
The graph of the piecewise function \( k(x) \) is divided into three distinct parts, corresponding to different intervals for \( x \):
1. **For \( x \leq -1 \):**
- The function \( k(x) \) is given by \( 3x + 3 \).
- This part of the graph is a straight line with a slope of 3 and a y-intercept at 3. On the graph, this is represented by a line segment extending to the left from the point \((-1, 0)\).
2. **For \( -1 < x \leq 1 \):**
- The function \( k(x) \) is given by \( 3x^2 - 2 \).
- This part of the graph is a parabola opening upwards. It intersects the y-axis at \((0, -2)\). Moving from left to right, it starts at \((-1, 1)\), reaches the vertex at \((0, -2)\), and continues to \((1, 1)\).
3. **For \( x > 1 \):**
- The function \( k(x) \) is given by \( 3 - x^2 \).
- This part of the graph is a downward-opening parabola. Starting from the point \((1, 2)\), it extends to the right, decreasing as \( x \) increases.
#### Key Points on the Graph:
- There is an open circle at the point \((-1, 0)\) indicating that this point is not included in the \( 3x + 3 \) segment.
- Another open circle is at the point \((1, 2)\) where the parabola \( 3 - x^2 \) starts but does not include this point.
By carefully analyzing these segments, one can understand the behavior of the piecewise function
![## Understanding Quadratic Functions Using Graphs
### Graphs of Quadratic Functions
In this image, we have two graphs of quadratic functions. Quadratic functions are typically represented in the form \( y = ax^2 + bx + c \).
#### Graph Analysis (Left Side)
- **Axes**: The graph has a vertical \(y\)-axis and a horizontal \(x\)-axis.
- **Parabola**: The parabola opens downwards, indicating that the coefficient of \( x^2 \) is negative (i.e., \( a < 0 \)).
- **X-Intercepts**: The parabola intersects the \( x \)-axis at approximately \( x = -2 \) and \( x = 2 \). These points represent the roots or solutions of the quadratic equation.
- **Y-Intercept**: It appears that there is no explicit \( y \)-intercept shown in this portion of the graph.
- **Vertex**: The highest point on the parabola (the vertex) is at \( x = 0 \), and it appears to be at about \( y = 4 \). This point is known as the maximum value of the quadratic function.
#### Graph Analysis (Right Side)
- **Axes**: The graph has a vertical \(y\)-axis and a horizontal \(x\)-axis.
- **Parabola**: The parabola opens upwards, indicating that the coefficient of \( x^2 \) is positive (i.e., \( a > 0 \)).
- **X-Intercepts**: The parabola intersects the \( x \)-axis at approximately \( x = -2 \) and \( x = 2 \). These points represent the roots or solutions of the quadratic equation.
- **Y-Intercept**: As with the previous graph, the \( y \)-intercept is not explicitly labeled.
- **Vertex**: The lowest point on the parabola (the vertex) is at \( x = 0 \), and it appears to be at about \( y = -4 \). This point is the minimum value of the quadratic function.
### Key Observations:
- **Symmetry**: Both graphs exhibit symmetry about the \( y \)-axis since the vertex is centered on the \( y \)-axis (i.e., at \( x = 0 \)).
- **Intercepts**: The intercepts on the \( x \)-axis indicate real](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86020f22-4efe-437b-8b05-308770976ff7%2F57a8960a-ca37-433b-916b-a6333aafda2c%2Fjoqduws_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding Quadratic Functions Using Graphs
### Graphs of Quadratic Functions
In this image, we have two graphs of quadratic functions. Quadratic functions are typically represented in the form \( y = ax^2 + bx + c \).
#### Graph Analysis (Left Side)
- **Axes**: The graph has a vertical \(y\)-axis and a horizontal \(x\)-axis.
- **Parabola**: The parabola opens downwards, indicating that the coefficient of \( x^2 \) is negative (i.e., \( a < 0 \)).
- **X-Intercepts**: The parabola intersects the \( x \)-axis at approximately \( x = -2 \) and \( x = 2 \). These points represent the roots or solutions of the quadratic equation.
- **Y-Intercept**: It appears that there is no explicit \( y \)-intercept shown in this portion of the graph.
- **Vertex**: The highest point on the parabola (the vertex) is at \( x = 0 \), and it appears to be at about \( y = 4 \). This point is known as the maximum value of the quadratic function.
#### Graph Analysis (Right Side)
- **Axes**: The graph has a vertical \(y\)-axis and a horizontal \(x\)-axis.
- **Parabola**: The parabola opens upwards, indicating that the coefficient of \( x^2 \) is positive (i.e., \( a > 0 \)).
- **X-Intercepts**: The parabola intersects the \( x \)-axis at approximately \( x = -2 \) and \( x = 2 \). These points represent the roots or solutions of the quadratic equation.
- **Y-Intercept**: As with the previous graph, the \( y \)-intercept is not explicitly labeled.
- **Vertex**: The lowest point on the parabola (the vertex) is at \( x = 0 \), and it appears to be at about \( y = -4 \). This point is the minimum value of the quadratic function.
### Key Observations:
- **Symmetry**: Both graphs exhibit symmetry about the \( y \)-axis since the vertex is centered on the \( y \)-axis (i.e., at \( x = 0 \)).
- **Intercepts**: The intercepts on the \( x \)-axis indicate real
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