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 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5dfe42cb-ec77-4eca-b109-21d7b9d5f896%2F1af233fe-cb42-45da-8f00-0c2cb8d546b8%2Fm5iu2c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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