Explanation There are a number of ways to write the quadratic equation a x 2 + b x + c . The value of a defines the opening shape of the function. 1. If a function is written in the standard form - a x 2 + b x + c , then the y-intercept is given by c . 2. If a function is written in the factored form - f ( x ) = a ( x - r ) ( x - s ) , then x-intercept is given by r and s . 3. If function is written in the vertex form - y = a ( x - h ) 2 + k , then the coordinates of the vertex are given by ( h , k ) . If the value of a is positive, the minimum value of the function is at k and if the value of a is negative, then the maximum value of function is at k . Given: f ( x ) is given as the square of the distance from point ( 2 , 1 ) to a point ( x , 3 x + 2 ) on the line y = 3 x + 2 Formula Used: 1. Distance between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given using formula ( x 1 – x 2 ) 2 + ( y 1 − y 2 ) 2 . 2. Rewrite the quadratic function in vertex form in order to find the maximum value of the function. Calculation: Two points are given as ( 2 , 1 ) and ( x , 3 x + 2 ) Distance between ( 2 , 1 ) and ( x , 3 x + 2 ) is given as ( x – 2 ) 2 + ( 3 x + 2 - 1 ) 2 ( x – 2 ) 2 + ( 3 x + 1 ) 2 Now, f ( x ) is given as the square of the distance from point ( 2 , 1 ) to a point ( x , 3 x + 2 ) Thus, f ( x ) = [ ( x – 2 ) 2 + ( 3 x + 1 ) 2 ]

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 1, Problem 31CRE

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The completion of the square of the function f(x), where f(x) is given as the square of the distance from point (2, 1) to a point (x, 3x + 2) on the line y = 3x + 2 and also find the minimum value of the function f(x).

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Chapter 1, Problem 30CRE

Chapter 1, Problem 32CRE

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Chapter 1 Solutions

Calculus - Standalone book

Ch. 1.1 - Prob. 1PQ Ch. 1.1 - Prob. 2PQ Ch. 1.1 - Prob. 3PQ Ch. 1.1 - Prob. 4PQ Ch. 1.1 - Prob. 5PQ Ch. 1.1 - Prob. 6PQ Ch. 1.1 - Prob. 7PQ Ch. 1.1 - Prob. 8PQ Ch. 1.1 - Prob. 9PQ Ch. 1.1 - Prob. 10PQ

Ch. 1.1 - Prob. 1E Ch. 1.1 - Prob. 2E Ch. 1.1 - Prob. 3E Ch. 1.1 - Prob. 4E Ch. 1.1 - Prob. 5E Ch. 1.1 - Prob. 6E Ch. 1.1 - Prob. 7E Ch. 1.1 - Prob. 8E Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E Ch. 1.1 - Prob. 11E Ch. 1.1 - Prob. 12E Ch. 1.1 - Prob. 13E Ch. 1.1 - Prob. 14E Ch. 1.1 - Prob. 15E Ch. 1.1 - Prob. 16E Ch. 1.1 - Prob. 17E Ch. 1.1 - Prob. 18E Ch. 1.1 - Prob. 19E Ch. 1.1 - Prob. 20E Ch. 1.1 - Prob. 21E Ch. 1.1 - Prob. 22E Ch. 1.1 - Prob. 23E Ch. 1.1 - Prob. 24E Ch. 1.1 - Prob. 25E Ch. 1.1 - Prob. 26E Ch. 1.1 - Prob. 27E Ch. 1.1 - Prob. 28E Ch. 1.1 - Prob. 29E Ch. 1.1 - Prob. 30E Ch. 1.1 - Prob. 31E Ch. 1.1 - Prob. 32E Ch. 1.1 - Prob. 33E Ch. 1.1 - Prob. 34E Ch. 1.1 - Prob. 35E Ch. 1.1 - Prob. 36E Ch. 1.1 - Prob. 37E Ch. 1.1 - Prob. 38E Ch. 1.1 - Prob. 39E Ch. 1.1 - Prob. 40E Ch. 1.1 - Prob. 41E Ch. 1.1 - Prob. 42E Ch. 1.1 - Prob. 43E Ch. 1.1 - Prob. 44E Ch. 1.1 - Prob. 45E Ch. 1.1 - Prob. 46E Ch. 1.1 - Prob. 47E Ch. 1.1 - Prob. 48E Ch. 1.1 - Prob. 49E Ch. 1.1 - Prob. 50E Ch. 1.1 - 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The completion of the square of the function f ( x ) , where f ( x ) is given as the square of the distance from point ( 2 , 1 ) to a point ( x , 3 x + 2 ) on the line y = 3 x + 2 and also find the minimum value of the function f ( x ) .

Solution Summary: The author explains how the quadratic equation a defines the opening shape of the function.

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The completion of the square of the function f(x), where f(x) is given as the square of the distance from point (2, 1) to a point (x, 3x + 2) on the line y = 3x + 2 and also find the minimum value of the function f(x).

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Chapter 1 Solutions