List 6 properties of the following quadratic function.be very specific,giving order pairs and interval answers where appropriate.Number each of six answers. ### Problem Statement**Task:** List 6 (six) properties of the following quadratic function. Ensure to number each of your six answers.The function given is:\[ f(x) = 4.5x^2 - 4.2x - 1 \]### Properties of the Quadratic FunctionWhen analyzing a quadratic function of the form $ ax^2 + bx + c $, several properties can be considered. Here are six possible properties to list:1. **Vertex:** The vertex form of a quadratic function can be determined by completing the square or using the vertex formula. For $ ax^2 + bx + c $, the vertex $(h,k)$ can be found using:\[ h = -\frac{b}{2a} \]\[ k = f(h) \]2. **Axis of Symmetry:** The vertical line that passes through the vertex and divides the parabola into two symmetrical halves is the axis of symmetry. For the quadratic function $ ax^2 + bx + c $, it is given by:\[ x = -\frac{b}{2a} \]3. **Direction of Opening:** The direction in which the parabola opens (upwards or downwards) depends on the coefficient of $ x^2 $ (i.e., the value of $ a $). If $ a > 0 $, the parabola opens upwards. If $ a < 0 $, it opens downwards.4. **Y-intercept:** The y-intercept of the quadratic function is the point where the graph intersects the y-axis, which occurs when $ x = 0 $. For the given quadratic function, the y-intercept is $ c $.5. **X-intercepts (Roots):** The x-intercepts are the points where the graph of the quadratic function intersects the x-axis. These occur when $ f(x) = 0 $. The roots can be found using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]6. **Discriminant:** The discriminant of the quadratic function provides information about the nature of the roots (real and distinct, real and repeated, or complex). The discriminant ($ \Delta $) is given by:\[ \Delta = b^2 - 4ac \]- If \( \Delta > 0 \

Given that ,List six properties.

Answered: List 6 (six) properties of the…

List 6 (six) properties of the following quadratic appropriate. Number each of your six answers 2 f(x) = 4.5x - 4.2x - 1

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

List 6 properties of the following quadratic function.be very specific,giving order pairs and interval answers where appropriate.Number each of six answers.

### Problem Statement
**Task:** List 6 (six) properties of the following quadratic function. Ensure to number each of your six answers.

The function given is:
\[ f(x) = 4.5x^2 - 4.2x - 1 \]

### Properties of the Quadratic Function
When analyzing a quadratic function of the form $ ax^2 + bx + c $, several properties can be considered. Here are six possible properties to list:

1. **Vertex:** The vertex form of a quadratic function can be determined by completing the square or using the vertex formula. For $ ax^2 + bx + c $, the vertex $(h,k)$ can be found using:
\[ h = -\frac{b}{2a} \]
\[ k = f(h) \]

2. **Axis of Symmetry:** The vertical line that passes through the vertex and divides the parabola into two symmetrical halves is the axis of symmetry. For the quadratic function $ ax^2 + bx + c $, it is given by:
\[ x = -\frac{b}{2a} \]

3. **Direction of Opening:** The direction in which the parabola opens (upwards or downwards) depends on the coefficient of $ x^2 $ (i.e., the value of $ a $). If $ a > 0 $, the parabola opens upwards. If $ a < 0 $, it opens downwards.

4. **Y-intercept:** The y-intercept of the quadratic function is the point where the graph intersects the y-axis, which occurs when $ x = 0 $. For the given quadratic function, the y-intercept is $ c $.

5. **X-intercepts (Roots):** The x-intercepts are the points where the graph of the quadratic function intersects the x-axis. These occur when $ f(x) = 0 $. The roots can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

6. **Discriminant:** The discriminant of the quadratic function provides information about the nature of the roots (real and distinct, real and repeated, or complex). The discriminant ($ \Delta $) is given by:
\[ \Delta = b^2 - 4ac \]
- If \( \Delta > 0 \

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