a. Answer The vertex of graph of function is ( − 1 , 9 ) . Explanation Given information: The given function is f ( x ) = 8 − 2 x − x 2 . Calculation: The given function can be written as, y = 8 − 2 x − x 2 .... ( Replace f ( x ) = y ) or, y − 8 = − ( x 2 + 2 x ) y − 8 + ( − 1 ) = − ( x 2 + 2 ( x ) ( 1 ) + ( 1 ) 2 ) .... ( Complete square ) y − 9 = − ( x + 1 ) 2 .... ( 1 ) Since (1) represents a parabola of the form y − k = a ( x − h ) 2 . Therefore the graph for the function has vertex ( − 1 , 9 ) . b. To determine The domain of the function f ( x ) = 8 − 2 x − x 2 . Answer The domain of the function is all real numbers. Explanation Given information: The given function is f ( x ) = 8 − 2 x − x 2 . Calculation: From (1), the parabola obtained represents a parabola opening downwards because of negative coefficient of ( x + 1 ) 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 ) = 8 − 2 x − x 2 . Answer The range of the function is { y | y ≤ 9 } . Explanation Given information: The given function is f ( x ) = 8 − 2 x − x 2 . Calculation: From (1), the parabola obtained represents a parabola opening downwards because of negative coefficient of ( x + 1 ) 2 . Also the parabola has the vertex ( − 1 , 9 ) which means the upper limit for range is 9 . Since, the parabola (1) is upper bound. So its maximum range will be ( y ) coordinate of its vertex. Hence the range of the function is all real numbers less than or equal to 9 . Mathematically, the range of function is { y | y ≤ 9 } . d. To determine The zeros of the function f ( x ) = 8 − 2 x − x 2 . Answer The zeros of the function are 2 , − 4 . Explanation Given information: The given function is f ( x ) = 8 − 2 x − x 2 . Calculation: The zeros of the function can be obtained by equating the given function equal to zero. Therefore f ( x ) = 0 gives, 8 − 2 x − x 2 = 0 or, x 2 + 2 x − 8 = 0 x = − 2 ± ( 2 ) 2 − 4 ( 1 ) ( − 8 ) 2 ( 1 ) = − ( 2 ) ± 6 2 = 2 , − 4 Hence the zeros of the function are 2 , − 4 .

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 27WE

To determine

The vertex of the graph of the function f(x)=8−2x−x2 .

Expert Solution

Answer to Problem 27WE

The vertex of graph of function is (−1,9) .

Explanation of Solution

Given information: The given function is f(x)=8−2x−x2 .

Calculation: The given function can be written as,

y=8−2x−x2.... (Replace f(x)=y)or, y−8=−(x2+2x)y−8+(−1)=−(x2+2(x)(1)+(1)2)....(Complete square)y−9=−(x+1)2 ....(1)

Since (1) represents a parabola of the form y−k=a(x−h)2 . Therefore the graph for the function has vertex (−1,9) .

To determine

The domain of the function f(x)=8−2x−x2 .

Expert Solution

Answer to Problem 27WE

The domain of the function is all real numbers.

Explanation of Solution

Given information: The given function is f(x)=8−2x−x2 .

Calculation: From (1), the parabola obtained represents a parabola opening downwards because of negative coefficient of (x+1)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)=8−2x−x2 .

Expert Solution

Answer to Problem 27WE

The range of the function is {y|y≤9} .

Explanation of Solution

Given information: The given function is f(x)=8−2x−x2 .

Calculation: From (1), the parabola obtained represents a parabola opening downwards because of negative coefficient of (x+1)2 . Also the parabola has the vertex (−1,9) which means the upper limit for range is 9 . Since, the parabola (1) is upper bound. So its maximum range will be (y) coordinate of its vertex.

Hence the range of the function is all real numbers less than or equal to 9 . Mathematically, the range of function is {y|y≤9} .

To determine

The zeros of the function f(x)=8−2x−x2 .

Expert Solution

Answer to Problem 27WE

The zeros of the function are 2,−4 .

Explanation of Solution

Given information: The given function is f(x)=8−2x−x2 .

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

8−2x−x2=0or, x2+2x−8=0x=−2±(2)2−4(1)(−8)2(1)=−(2)±62=2,−4

Hence the zeros of the function are 2,−4 .

Chapter 7.6, Problem 26WE

Chapter 7.6, Problem 28WE

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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