a. Answer The vertex of graph of function is ( 2 , − 7 ) . Explanation Given information: The given function is f ( x ) = x 2 − 4 x − 3 . Calculation: The given function can be written as, y = x 2 − 4 x − 3.... ( Replace f ( x ) = y ) or, y + 3 = x 2 − 4 x y + 3 + ( 4 ) = ( x 2 − 2 ( x ) ( 2 ) + ( 2 ) 2 ) .... ( Complete square ) y + 7 = ( x − 2 ) 2 .... ( 1 ) Since (1) represents a parabola of the form y − k = a ( x − h ) 2 . Therefore the graph for the function has vertex ( 2 , − 7 ) . b. To determine The domain of the function f ( x ) = x 2 − 4 x − 3 . Answer The domain of the function is all real numbers. Explanation Given information: The given function is f ( x ) = x 2 − 4 x − 3 . Calculation: From (1), the parabola obtained is y + 7 = ( x − 2 ) 2 which represents a parabola opening upwards because of positive coefficient of ( x − 2 ) 2 . Therefore the domain of the function has no restrictions at all. Hence the domain of the function is all real numbers. c. To determine The range of the function f ( x ) = x 2 − 4 x − 3 . Answer The range of the function is { y | y ≥ − 7 } . Explanation Given information: The given function is f ( x ) = x 2 − 4 x − 3 . Calculation: From (1), the parabola obtained is y + 7 = ( x − 2 ) 2 which represents a parabola opening upwards because of positive coefficient of ( x − 2 ) 2 . Also the parabola has the vertex ( 2 , − 7 ) which means the lower limit for range is − 7 . Since, the parabola (1) is not upper bound. Hence the range of the function is all real numbers greater than or equal to − 7 . Mathematically, the range of function is { y | y ≥ − 7 } . d. To determine The zeros of the function f ( x ) = x 2 − 4 x − 3 . Answer The zeros of the function are 2 + 7 , 2 − 7 . Explanation Given information: The given function is f ( x ) = x 2 − 4 x − 3 . Calculation: The zeros of the function can be obtained by equating the given function equal to zero. Therefore f ( x ) = 0 gives, x 2 − 4 x − 3 = 0 x = − ( − 4 ) ± ( − 4 ) 2 − 4 ( 1 ) ( − 3 ) 2 ( 1 ) .... ( Quadratic formula ) = 4 ± 28 2 = 4 ± 2 7 2 = 2 + 7 , 2 − 7 Hence the zeros of the function are 2 + 7 , 2 − 7 .

Algebra and Trigonometry: Structure and Method, Book 2

2000th Edition

ISBN: 9780395977255

Author: MCDOUGAL LITTEL

Publisher: McDougal Littell

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Question

Chapter 7.6, Problem 25WE

To determine

The vertex of the graph of the function f(x)=x2−4x−3 .

Expert Solution

Answer to Problem 25WE

The vertex of graph of function is (2,−7) .

Explanation of Solution

Given information: The given function is f(x)=x2−4x−3 .

Calculation: The given function can be written as,

y=x2−4x−3.... (Replace f(x)=y)or, y+3=x2−4xy+3+(4)=(x2−2(x)(2)+(2)2)....(Complete square)y+7=(x−2)2 ....(1)

Since (1) represents a parabola of the form y−k=a(x−h)2 . Therefore the graph for the function has vertex (2,−7) .

To determine

The domain of the function f(x)=x2−4x−3 .

Expert Solution

Answer to Problem 25WE

The domain of the function is all real numbers.

Explanation of Solution

Given information: The given function is f(x)=x2−4x−3 .

Calculation: From (1), the parabola obtained is y+7=(x−2)2 which represents a parabola opening upwards because of positive coefficient of (x−2)2 . Therefore the domain of the function has no restrictions at all.

Hence the domain of the function is all real numbers.

To determine

The range of the function f(x)=x2−4x−3 .

Expert Solution

Answer to Problem 25WE

The range of the function is {y|y≥−7} .

Explanation of Solution

Given information: The given function is f(x)=x2−4x−3 .

Calculation: From (1), the parabola obtained is y+7=(x−2)2 which represents a parabola opening upwards because of positive coefficient of (x−2)2 . Also the parabola has the vertex (2,−7) which means the lower limit for range is −7 . Since, the parabola (1) is not upper bound.

Hence the range of the function is all real numbers greater than or equal to −7 . Mathematically, the range of function is {y|y≥−7} .

To determine

The zeros of the function f(x)=x2−4x−3 .

Expert Solution

Answer to Problem 25WE

The zeros of the function are 2+7,2−7 .

Explanation of Solution

Given information: The given function is f(x)=x2−4x−3 .

Calculation: The zeros of the function can be obtained by equating the given function equal to zero. Therefore f(x)=0 gives,

x2−4x−3=0x=−(−4)±(−4)2−4(1)(−3)2(1)....(Quadratic formula)=4±282=4±272=2+7,2−7

Hence the zeros of the function are 2+7,2−7 .

Chapter 7.6, Problem 24WE

Chapter 7.6, Problem 26WE

Chapter 7 Solutions

Algebra and Trigonometry: Structure and Method, Book 2

Ch. 7.1 - Prob. 1OE Ch. 7.1 - Prob. 2OE Ch. 7.1 - Prob. 3OE Ch. 7.1 - Prob. 4OE Ch. 7.1 - Prob. 5OE Ch. 7.1 - Prob. 6OE Ch. 7.1 - Prob. 7OE Ch. 7.1 - Prob. 8OE Ch. 7.1 - Prob. 9OE Ch. 7.1 - Prob. 10OE

Ch. 7.1 - Prob. 11OE Ch. 7.1 - Prob. 12OE Ch. 7.1 - Prob. 13OE Ch. 7.1 - Prob. 14OE Ch. 7.1 - Prob. 15OE Ch. 7.1 - Prob. 16OE Ch. 7.1 - Prob. 17OE Ch. 7.1 - Prob. 18OE Ch. 7.1 - Prob. 19OE Ch. 7.1 - Prob. 20OE Ch. 7.1 - Prob. 1WE Ch. 7.1 - Prob. 2WE Ch. 7.1 - Prob. 3WE Ch. 7.1 - Prob. 4WE Ch. 7.1 - Prob. 5WE Ch. 7.1 - Prob. 6WE Ch. 7.1 - Prob. 7WE Ch. 7.1 - Prob. 8WE Ch. 7.1 - Prob. 9WE Ch. 7.1 - Prob. 10WE Ch. 7.1 - Prob. 11WE Ch. 7.1 - Prob. 12WE Ch. 7.1 - Prob. 13WE Ch. 7.1 - Prob. 14WE Ch. 7.1 - Prob. 15WE Ch. 7.1 - Prob. 16WE Ch. 7.1 - Prob. 17WE Ch. 7.1 - Prob. 18WE Ch. 7.1 - Prob. 19WE Ch. 7.1 - Prob. 20WE Ch. 7.1 - Prob. 21WE Ch. 7.1 - Prob. 22WE Ch. 7.1 - Prob. 23WE Ch. 7.1 - Prob. 24WE Ch. 7.1 - Prob. 25WE Ch. 7.1 - Prob. 26WE Ch. 7.1 - Prob. 27WE Ch. 7.1 - Prob. 28WE Ch. 7.1 - Prob. 29WE Ch. 7.1 - Prob. 30WE Ch. 7.1 - Prob. 31WE Ch. 7.1 - Prob. 32WE Ch. 7.1 - Prob. 33WE Ch. 7.1 - Prob. 34WE Ch. 7.1 - Prob. 35WE Ch. 7.1 - Prob. 36WE Ch. 7.1 - Prob. 37WE Ch. 7.1 - Prob. 38WE Ch. 7.1 - 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