a. Answer The vertex of graph of function is ( 1 , − 1 2 ) . Explanation Given information: The given function is H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Calculation: The given function can be written as, y = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) .... ( Replace H ( x ) = y ) or, y = 1 2 ( 9 x 2 − 18 x + 8 ) y = 9 2 ( x 2 − 2 x + 8 9 ) y + ( 1 2 ) = 9 2 ( x 2 − 2 ( x ) ( 1 ) + ( 1 ) 2 ) .... ( Complete square ) y + 1 2 = ( 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 , − 1 2 ) . b. To determine The domain of the function H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Answer The domain of the function is all real numbers. Explanation Given information: The given function is H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Calculation: From (1), the parabola obtained represents a parabola opening upwards because of positive 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 H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Answer The range of the function is { y | y ≥ − 1 2 } . Explanation Given information: The given function is H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Calculation: From (1) the parabola obtained represents a parabola opening upwards because of positive coefficient of ( x − 1 ) 2 . Also the parabola has the vertex ( 1 , − 1 2 ) which means the lower limit for range is − 1 2 . Since, the parabola (1) is lower bound, therefore the range of the function is all real numbers greater than or equal to − 1 2 . Mathematically, the range of function is { y | y ≥ − 1 2 } . d. To determine The zeros of the function H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Answer The zeros of the function are 4 3 , 2 3 . Explanation Given information: The given function is H ( x ) = 1 2 ( 3 x − 2 ) ( 3 x − 4 ) . Calculation: The zeros of the function can be obtained by equating the given function equal to zero. Therefore H ( x ) = 0 gives, 1 2 ( 3 x − 2 ) ( 3 x − 4 ) = 0 ( 3 x − 2 ) ( 3 x − 4 ) = 0 3 x − 2 = 0 or 3 x − 4 = 0 x = 2 3 or x = 4 3 Hence the zeros of the function are 4 3 , 2 3 .

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 36WE

To determine

The vertex of the graph of the function H(x)=12(3x−2)(3x−4) .

Expert Solution

Answer to Problem 36WE

The vertex of graph of function is (1,−12) .

Explanation of Solution

Given information: The given function is H(x)=12(3x−2)(3x−4) .

Calculation: The given function can be written as,

y=12(3x−2)(3x−4).... (Replace H(x)=y)or, y=12(9x2−18x+8)y=92(x2−2x+89)y+(12)=92(x2−2(x)(1)+(1)2)....(Complete square)y+12=(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,−12) .

To determine

The domain of the function H(x)=12(3x−2)(3x−4) .

Expert Solution

Answer to Problem 36WE

The domain of the function is all real numbers.

Explanation of Solution

Given information: The given function is H(x)=12(3x−2)(3x−4) .

Calculation: From (1), the parabola obtained represents a parabola opening upwards because of positive 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 H(x)=12(3x−2)(3x−4) .

Expert Solution

Answer to Problem 36WE

The range of the function is {y|y≥−12} .

Explanation of Solution

Given information: The given function is H(x)=12(3x−2)(3x−4) .

Calculation: From (1) the parabola obtained represents a parabola opening upwards because of positive coefficient of (x−1)2 . Also the parabola has the vertex (1,−12) which means the lower limit for range is −12 . Since, the parabola (1) is lower bound, therefore the range of the function is all real numbers greater than or equal to −12 . Mathematically, the range of function is {y|y≥−12} .

To determine

The zeros of the function H(x)=12(3x−2)(3x−4) .

Expert Solution

Answer to Problem 36WE

The zeros of the function are 43,23 .

Explanation of Solution

Given information: The given function is H(x)=12(3x−2)(3x−4) .

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

12(3x−2)(3x−4)=0(3x−2)(3x−4)=03x−2=0 or 3x−4=0x=23 or x=43

Hence the zeros of the function are 43,23 .

Chapter 7.6, Problem 35WE

Chapter 7.6, Problem 37WE

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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