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Sketch the graph of the function. (3x + 3, X ≤ -1 k(x) = 3x²-2, -1 1 y

Sketch the graph of the function. (3x + 3, X ≤ -1 k(x) = 3x²-2, -1 1 y

Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter8: Introduction To Functions
Section8.8: Linear And Quadratic Functions
Problem 2WE
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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
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
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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