### Quadratic Functions and Their PropertiesGiven 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-InterceptTo 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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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)

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

### 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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