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15. f(x) = x² - 6x – 16 %3D

15. f(x) = x² - 6x – 16 %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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Question

1) Find the vertex of each parabola. 2) Use the discriminant to determine the number of x-intercepts the graph will have. Then determine the x-intercepts. 

### Quadratic Function Analysis **Problem 15:** Given the function \( f(x) = x^2 - 6x - 16 \). This is a quadratic function in the standard form \( f(x) = ax^2 + bx + c \), where: - \( a = 1 \) - \( b = -6 \) - \( c = -16 \) ### Key Concepts - **Parabolic Shape:** The function represents a parabola opening upwards, as the coefficient of \( x^2 \) (which is \( a \)) is positive. - **Vertex Form:** Transforming this into vertex form can help find its vertex and axis of symmetry. - **Root Calculation:** Using the quadratic formula or factoring, find the roots of the equation \( x^2 - 6x - 16 = 0 \) for \( f(x) = 0 \). - **Graph Details:** The vertex's \( x \)-coordinate is found using \( x = -\frac{b}{2a} \). The function's axis of symmetry is the vertical line passing through this \( x \)-coordinate. For further exploration, graph this equation to visually understand the changes in behavior across different \( x \)-values. Analyzing the vertex and intercepts provides insight into solving real-world quadratic problems.
Transcribed Image Text:### Quadratic Function Analysis **Problem 15:** Given the function \( f(x) = x^2 - 6x - 16 \). This is a quadratic function in the standard form \( f(x) = ax^2 + bx + c \), where: - \( a = 1 \) - \( b = -6 \) - \( c = -16 \) ### Key Concepts - **Parabolic Shape:** The function represents a parabola opening upwards, as the coefficient of \( x^2 \) (which is \( a \)) is positive. - **Vertex Form:** Transforming this into vertex form can help find its vertex and axis of symmetry. - **Root Calculation:** Using the quadratic formula or factoring, find the roots of the equation \( x^2 - 6x - 16 = 0 \) for \( f(x) = 0 \). - **Graph Details:** The vertex's \( x \)-coordinate is found using \( x = -\frac{b}{2a} \). The function's axis of symmetry is the vertical line passing through this \( x \)-coordinate. For further exploration, graph this equation to visually understand the changes in behavior across different \( x \)-values. Analyzing the vertex and intercepts provides insight into solving real-world quadratic problems.
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