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3 Express each function in the form a(x + b)2 +c, where a b and c are constants and hence state the range of each function. a f(x) = x² + 4x - 1 for x = R

3 Express each function in the form a(x + b)2 +c, where a b and c are constants and hence state the range of each function. a f(x) = x² + 4x - 1 for x = R

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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Please help me answer questions 3.A, 4.A, 5, 6, and 7

**Mathematical Functions and Range Exercises** 1. **Express Each Function and Determine the Range** Consider each function in the form \( a(x + b)^2 + c \), where \( a, b, \) and \( c \) are constants. Determine the range of each function. a. \( f(x) = x^2 + 4x - 1 \) for \( x \in \mathbb{R} \) b. \( f(x) = 2x^2 - 4x + 3 \) for \( x \in \mathbb{R} \) 2. **Alternate Expression and Range Determination** Express each function in the form \( a - b(x + c)^2 \), where \( a, b, \) and \( c \) are constants. Determine the range of each. a. \( f(x) = 3 - 2x - x^2 \) for \( x \in \mathbb{R} \) b. \( f(x) = 1 - 6x - x^2 \) for \( x \in \mathbb{R} \) 3. **Function and Range Calculation** The function \( g: x \mapsto 6 + 3ax - 3x^2 \), where \( a \) is a constant, is defined for \( x \in \mathbb{R} \). Find the range of \( g \) in terms of \( a \). 4. **Select Range for \( f(x) = x^2 - 4 \)** For \( x \in \mathbb{R} \), \(-a \leq x \leq a\). If the range of \( f \) is \(-4 \leq f(x) \leq 5\), find the value of \( a \). 5. **Specific Function Transformations** Consider \( f(x) = 4x^2 - 8x + 2 \) for \( x \in \mathbb{R} \), \( 0 \leq x \leq k \). a. Express \( f(x) \) in the form \( a(x + b)^2 + c \). b. Determine the value of \( k \) for which the graph of \( y = f(x
Transcribed Image Text:**Mathematical Functions and Range Exercises** 1. **Express Each Function and Determine the Range** Consider each function in the form \( a(x + b)^2 + c \), where \( a, b, \) and \( c \) are constants. Determine the range of each function. a. \( f(x) = x^2 + 4x - 1 \) for \( x \in \mathbb{R} \) b. \( f(x) = 2x^2 - 4x + 3 \) for \( x \in \mathbb{R} \) 2. **Alternate Expression and Range Determination** Express each function in the form \( a - b(x + c)^2 \), where \( a, b, \) and \( c \) are constants. Determine the range of each. a. \( f(x) = 3 - 2x - x^2 \) for \( x \in \mathbb{R} \) b. \( f(x) = 1 - 6x - x^2 \) for \( x \in \mathbb{R} \) 3. **Function and Range Calculation** The function \( g: x \mapsto 6 + 3ax - 3x^2 \), where \( a \) is a constant, is defined for \( x \in \mathbb{R} \). Find the range of \( g \) in terms of \( a \). 4. **Select Range for \( f(x) = x^2 - 4 \)** For \( x \in \mathbb{R} \), \(-a \leq x \leq a\). If the range of \( f \) is \(-4 \leq f(x) \leq 5\), find the value of \( a \). 5. **Specific Function Transformations** Consider \( f(x) = 4x^2 - 8x + 2 \) for \( x \in \mathbb{R} \), \( 0 \leq x \leq k \). a. Express \( f(x) \) in the form \( a(x + b)^2 + c \). b. Determine the value of \( k \) for which the graph of \( y = f(x
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