a. Answer The vertex of graph of function is ( − 2 , − 3 ) . Explanation Given information: The given function is G ( x ) = 2 x 2 + 8 x + 5 . Calculation: The given function can be written as, y = 2 x 2 + 8 x + 5.... ( Replace G ( x ) = y ) or, y − 5 = 2 ( x 2 + 4 x ) or, y − 5 + ( 8 ) = 2 ( x 2 + 2 ( x ) ( 2 ) + ( 2 ) 2 ) ( Complete square ) y + 3 = 2 ( 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 , − 3 ) . b. To determine The domain of the function G ( x ) = 2 x 2 + 8 x + 5 . Answer The domain of the function is all real numbers. Explanation Given information: The given function is G ( x ) = 2 x 2 + 8 x + 5 . Calculation: From (1), the parabola obtained 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 G ( x ) = 2 x 2 + 8 x + 5 . Answer The range of the function is { y | y ≥ − 3 } . Explanation Given information: The given function is f ( x ) = x 2 − 2 x − 5 . Calculation: From (1) the parabola obtained represents a parabola opening upwards because of positive coefficient of ( x + 2 ) 2 . Also the parabola has the vertex ( − 2. − 3 ) which means the lower limit for range is − 3 . Since, the parabola (1) is lower bound, therefore the range of the function is all real numbers greater than or equal to − 3 . Mathematically, the range of function is { y | y ≥ − 3 } . d. To determine The zeros of the function G ( x ) = 2 x 2 + 8 x + 5 . Answer The zeros of the function are − 2 + 6 2 , − 2 − 6 2 . Explanation Given information: The given function is G ( x ) = 2 x 2 + 8 x + 5 . Calculation: The zeros of the function can be obtained by equating the given function equal to zero. Therefore G ( x ) = 0 gives, 2 x 2 + 8 x + 5 = 0 x = − ( 8 ) ± ( 8 ) 2 − 4 ( 2 ) ( 5 ) 2 ( 2 ) ( Quadratic formula ) x = − 8 ± 2 6 4 x = − 2 + 6 2 , − 2 − 6 2 Hence the zeros of the function are − 2 + 6 2 , − 2 − 6 2 .

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 30WE

To determine

The vertex of the graph of the function G(x)=2x2+8x+5 .

Expert Solution

Answer to Problem 30WE

The vertex of graph of function is (−2,−3) .

Explanation of Solution

Given information: The given function is G(x)=2x2+8x+5 .

Calculation: The given function can be written as,

y=2x2+8x+5.... (Replace G(x)=y)or, y−5=2(x2+4x)or, y−5+(8)=2(x2+2(x)(2)+(2)2) (Complete square)y+3=2(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,−3) .

To determine

The domain of the function G(x)=2x2+8x+5 .

Expert Solution

Answer to Problem 30WE

The domain of the function is all real numbers.

Explanation of Solution

Given information: The given function is G(x)=2x2+8x+5 .

Calculation: From (1), the parabola obtained 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 G(x)=2x2+8x+5 .

Expert Solution

Answer to Problem 30WE

The range of the function is {y|y≥−3} .

Explanation of Solution

Given information: The given function is f(x)=x2−2x−5 .

Calculation: From (1) the parabola obtained represents a parabola opening upwards because of positive coefficient of (x+2)2 . Also the parabola has the vertex (−2.−3) which means the lower limit for range is −3 . Since, the parabola (1) is lower bound, therefore the range of the function is all real numbers greater than or equal to −3 . Mathematically, the range of function is {y|y≥−3} .

To determine

The zeros of the function G(x)=2x2+8x+5 .

Expert Solution

Answer to Problem 30WE

The zeros of the function are −2+62,−2−62 .

Explanation of Solution

Given information: The given function is G(x)=2x2+8x+5 .

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

2x2+8x+5=0x=−(8)±(8)2−4(2)(5)2(2) (Quadratic formula)x=−8±264x=−2+62,−2−62

Hence the zeros of the function are −2+62,−2−62 .

Chapter 7.6, Problem 29WE

Chapter 7.6, Problem 31WE

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

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