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Determine the domain and range of the quadratic function. f (x) = –2(x + 4)² – 6 -

Determine the domain and range of the quadratic function. f (x) = –2(x + 4)² – 6 -

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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**Determine the Domain and Range of the Quadratic Function** Given the quadratic function: \[ f(x) = -2(x + 4)^2 - 6 \] **Domain:** The domain of any quadratic function, including this one, is the set of all real numbers. This is because you can substitute any real number for \( x \) without encountering any restrictions like division by zero or non-real numbers. **Range:** To determine the range, we need to analyze the function. The quadratic expression \( (x + 4)^2 \) represents a parabola that opens upwards. However, due to the negative coefficient \(-2\), the parabola opens downwards. The vertex form of a parabola is given as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex. In this function, \( h = -4 \) and \( k = -6 \). Since the parabola opens downwards, the vertex represents the maximum point. Therefore, the \( y \)-value at the vertex is the maximum value of the function. - The maximum value of the function is \( -6 \). Thus, the range of the function is all real numbers \( y \) such that \( y \leq -6 \).
Transcribed Image Text:**Determine the Domain and Range of the Quadratic Function** Given the quadratic function: \[ f(x) = -2(x + 4)^2 - 6 \] **Domain:** The domain of any quadratic function, including this one, is the set of all real numbers. This is because you can substitute any real number for \( x \) without encountering any restrictions like division by zero or non-real numbers. **Range:** To determine the range, we need to analyze the function. The quadratic expression \( (x + 4)^2 \) represents a parabola that opens upwards. However, due to the negative coefficient \(-2\), the parabola opens downwards. The vertex form of a parabola is given as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex. In this function, \( h = -4 \) and \( k = -6 \). Since the parabola opens downwards, the vertex represents the maximum point. Therefore, the \( y \)-value at the vertex is the maximum value of the function. - The maximum value of the function is \( -6 \). Thus, the range of the function is all real numbers \( y \) such that \( y \leq -6 \).
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