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Glven y = ax + bx + chas an axis of symmetry at x = 3with an x intercept at (5, 0). a. Determine the other x intercept. ax + bx + c. b. Given y < 18 find the function in form y c. Determine where the function has a horizontal tangent line. %3D

Glven y = ax + bx + chas an axis of symmetry at x = 3with an x intercept at (5, 0). a. Determine the other x intercept. ax + bx + c. b. Given y < 18 find the function in form y c. Determine where the function has a horizontal tangent line. %3D

Algebra and Trigonometry (6th Edition)
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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### Quadratic Functions and Their Properties Given the quadratic function \( y = ax^2 + bx + c \), consider the following properties and tasks. This function has an axis of symmetry at \( x = 3 \) and an x-intercept at \( (5,0) \). #### a. Determine the Other X-Intercept To find the other x-intercept, let's use the property that the vertex form of a parabola reveals its axis of symmetry. Since the axis of symmetry is at \( x = 3 \), and the x-intercept is \( (5,0) \): 1. The axis of symmetry tells us that the vertex is equally distant from both x-intercepts. 2. Since the first intercept is at \( (5,0) \), we need to find the point that is symmetric about \( x = 3 \). This point will be \( (1,0) \). Thus, the other x-intercept is \( (1,0) \). #### b. Given \( y \leq 18 \), Find the Function in the Form \( y = ax^2 + bx + c \) 1. Start by using the known x-intercepts, \( (5,0) \) and \( (1,0) \), in the factored form of the quadratic equation: \[ y = a(x - 5)(x - 1) \] 2. Expanding this, we get: \[ y = a(x^2 - 6x + 5) \] 3. Since the maximum value \( y \leq 18 \) occurs at the vertex, let's find the vertex's y-coordinate. The vertex \( x \) value is at \( x = 3 \) because the axis of symmetry is \( x = 3 \): \[ y = a(3^2 - 6 \cdot 3 + 5) \] \[ y = a(9 - 18 + 5) \] \[ y = a(-4) \] 4. The maximum value is given as 18. Therefore, at the vertex: \[ -4a = 18 \] \[ a = -\frac{18}{4} = -\frac{9}{2} \] Thus, the function is: \[ y = -\frac{9}{2}(x^2 - 6x + 5) \] \[ y =
Transcribed Image Text:### Quadratic Functions and Their Properties Given the quadratic function \( y = ax^2 + bx + c \), consider the following properties and tasks. This function has an axis of symmetry at \( x = 3 \) and an x-intercept at \( (5,0) \). #### a. Determine the Other X-Intercept To find the other x-intercept, let's use the property that the vertex form of a parabola reveals its axis of symmetry. Since the axis of symmetry is at \( x = 3 \), and the x-intercept is \( (5,0) \): 1. The axis of symmetry tells us that the vertex is equally distant from both x-intercepts. 2. Since the first intercept is at \( (5,0) \), we need to find the point that is symmetric about \( x = 3 \). This point will be \( (1,0) \). Thus, the other x-intercept is \( (1,0) \). #### b. Given \( y \leq 18 \), Find the Function in the Form \( y = ax^2 + bx + c \) 1. Start by using the known x-intercepts, \( (5,0) \) and \( (1,0) \), in the factored form of the quadratic equation: \[ y = a(x - 5)(x - 1) \] 2. Expanding this, we get: \[ y = a(x^2 - 6x + 5) \] 3. Since the maximum value \( y \leq 18 \) occurs at the vertex, let's find the vertex's y-coordinate. The vertex \( x \) value is at \( x = 3 \) because the axis of symmetry is \( x = 3 \): \[ y = a(3^2 - 6 \cdot 3 + 5) \] \[ y = a(9 - 18 + 5) \] \[ y = a(-4) \] 4. The maximum value is given as 18. Therefore, at the vertex: \[ -4a = 18 \] \[ a = -\frac{18}{4} = -\frac{9}{2} \] Thus, the function is: \[ y = -\frac{9}{2}(x^2 - 6x + 5) \] \[ y =
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