(a) Explanation Given: f ( x ) = 4 x − 6 + 3 g ( x ) = 16 x + 1 − 6 h ( x ) = x + 3 + 2 Graph: (b) To determine Identify the transformation on the graph of the parent function.Find the values caused each transformation. Explanation Given: f ( x ) = 4 x − 6 + 3 g ( x ) = 16 x + 1 − 6 h ( x ) = x + 3 + 2 Calculation: The parent function is p ( x ) = x For f ( x ) = 4 x − 6 + 3 , the parent function is shifted 6 units to the right on the x-axis , then stretched vertically by a factor of 4 , then shifted 3 units upward on the y-axis. For g ( x ) = 16 x + 1 − 6 = 4 x + 1 16 − 6 , the parent function is shifted 1 16 units to the left on the x-axis , then stretched vertically by a factor of 4, then shifted 6 units downward on the y-axis. For h ( x ) = x + 3 + 2 , the parent function is shifted 3 units to the left on the x-axis , then shifted 2 units upward on the y-axis. (c) To determine Find the functions that appear to be stretched or compressed vertically. Answer f(x) and g(x) appear stretched vertically by a factor of 4 units. Explanation Given: f ( x ) = 4 x − 6 + 3 g ( x ) = 16 x + 1 − 6 h ( x ) = x + 3 + 2 Calculation: f ( x ) = 4 x − 6 + 3 = 16 ( x − 6 ) + 3 g ( x ) = 16 x + 1 − 6 = 16 ( x + 1 16 ) − 6 In both of the functions, since x is multiplied by a factor of 16 , they appear stretched vertically by a factor of 16 = 4 units. (d) To determine The two functions that are stretched appear to be stretched by the same magnitude.Explain how it is possible. Explanation Given: f ( x ) = 4 x − 6 + 3 g ( x ) = 16 x + 1 − 6 h ( x ) = x + 3 + 2 Calculation: In the function f (x), if you take 4 inside the square root , you will get f ( x ) = 4 x − 6 + 3 = 16 ( x − 6 ) + 3 Similarly , for g(x) , if you take 16 common , you will get g ( x ) = 16 x + 1 − 6 = 16 ( x + 1 16 ) − 6 So, in both of the functions, since x is multiplied by a factor of 16 , they appear stretched vertically by a factor of 16 = 4 units. (e) To determine Make a table of the rate of change for all three functions between 8 and 12 as compared to 12 and 16. Explain the generalization about rate of change in square root function that can be made as a result of your findings. Answer the rate of change of the square root a x − b + c functions decreases as the interval of x increases , where a > 0. Explanation Given: f ( x ) = 4 x − 6 + 3 g ( x ) = 16 x + 1 − 6 h ( x ) = x + 3 + 2 Calculation: Rate of change of a function y = m x + c on interval [ x 1 , x 2 ] is given by y 2 − y 1 x 2 − x 1 where y 1 = m x 1 + c and y 2 = m x 2 + c Function [8,12] [12,16] f ( x ) = y = 4 x − 6 + 3 4 12 − 6 + 3 − ( 4 8 − 6 + 3 ) 12 − 8 = 4 6 − 4 2 4 = 6 − 2 ≈ 1.03 4 16 − 6 + 3 − ( 4 12 − 6 + 3 ) 16 − 12 = 4 10 − 4 6 4 = 10 − 6 ≈ 0.71 g ( x ) = y = 16 x + 1 − 6 16 ( 12 ) + 1 − 6 − ( 16 ( 8 ) + 1 − 6 ) 12 − 8 = 193 − 129 4 ≈ 0.633 16 ( 16 ) + 1 − 6 − ( 16 ( 12 ) + 1 − 6 ) 16 − 12 = 257 − 193 4 ≈ 0.534 h ( x ) = y = x + 3 + 2 12 + 3 + 2 − ( 8 + 3 + 2 ) 12 − 8 = 15 − 11 4 ≈ 0.13 16 + 3 + 2 − ( 12 + 3 + 2 ) 16 − 12 = 19 − 15 4 ≈ 0.12 So, from the table , we observe that the rate of change of the square root a x − b + c functions decreases as the interval of x increases , where a > 0.

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Author: McGraw-Hill Glencoe

Publisher: MCG

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Question

Chapter 6.3, Problem 44PPS

(a)

To determine

Graph each function on the same set of axes.

(a)

Expert Solution

Explanation of Solution

Given:

$f(x)=4x−6+3g(x)=16x+1−6h(x)=x+3+2$

Graph:

Glencoe Algebra 2 Student Edition C2014, Chapter 6.3, Problem 44PPS

(b)

To determine

Identify the transformation on the graph of the parent function.Find the values caused each transformation.

(b)

Expert Solution

Explanation of Solution

Given:

$f(x)=4x−6+3g(x)=16x+1−6h(x)=x+3+2$

Calculation:

The parent function is $p(x)=x$

For $f(x)=4x−6+3$ , the parent function is shifted 6 units to the right on the x-axis , then stretched vertically by a factor of 4 , then shifted 3 units upward on the y-axis.

For $g(x)=16x+1−6=4x+116−6$ , the parent function is shifted $116$ units to the left on the x-axis , then stretched vertically by a factor of 4, then shifted 6 units downward on the y-axis.

For $h(x)=x+3+2$ , the parent function is shifted 3 units to the left on the x-axis , then shifted 2 units upward on the y-axis.

(c)

To determine

Find the functions that appear to be stretched or compressed vertically.

(c)

Expert Solution

Answer to Problem 44PPS

f(x) and g(x) appear stretched vertically by a factor of 4 units.

Explanation of Solution

Given:

$f(x)=4x−6+3g(x)=16x+1−6h(x)=x+3+2$

Calculation: $f(x)=4x−6+3=16(x−6)+3$

$g(x)=16x+1−6=16(x+116)−6$

In both of the functions, since x is multiplied by a factor of 16 , they appear stretched vertically by a factor of $16=4$ units.

(d)

To determine

The two functions that are stretched appear to be stretched by the same magnitude.Explain how it is possible.

(d)

Expert Solution

Explanation of Solution

Given:

$f(x)=4x−6+3g(x)=16x+1−6h(x)=x+3+2$

Calculation: In the function f (x), if you take 4 inside the square root , you will get $f(x)=4x−6+3=16(x−6)+3$

Similarly , for g(x) , if you take 16 common , you will get

$g(x)=16x+1−6=16(x+116)−6$

So, in both of the functions, since x is multiplied by a factor of 16 , they appear stretched vertically by a factor of $16=4$ units.

(e)

To determine

Make a table of the rate of change for all three functions between 8 and 12 as compared to 12 and 16. Explain the generalization about rate of change in square root function that can be made as a result of your findings.

(e)

Expert Solution

Answer to Problem 44PPS

the rate of change of the square root $ax−b+c$ functions decreases as the interval of x increases , where a > 0.

Explanation of Solution

Given:

$f(x)=4x−6+3g(x)=16x+1−6h(x)=x+3+2$

Calculation:

Rate of change of a function $y=mx+c$ on interval $[x1,x2]$ is given by $y2−y1x2−x1$ where $y1=mx1+c and y2=mx2+c$

Function	[8,12]	[12,16]
$f(x)=y=4x−6+3$	$4 12−6+3−( 4 8−6 +3)12−8 =46−424=6−2≈1.03$	$4 16−6+3−( 4 12−6 +3)16−12 =4 10−464=10−6≈0.71$
$g(x)=y=16x+1−6$	$16(12)+1−6−( 16(8)+1 −6)12−8 = 193− 1294≈0.633$	$16(16)+1−6−( 16(12)+1 −6)16−12 = 257− 1934≈0.534$
$h(x)=y=x+3+2$	$12+3+2−( 8+3 +2)12−8 = 15− 114≈0.13$	$16+3+2−( 12+3 +2)16−12 = 19− 154≈0.12$

So, from the table , we observe that the rate of change of the square root $ax−b+c$ functions decreases as the interval of x increases , where a > 0.

Chapter 6.3, Problem 43PPS

Chapter 6.3, Problem 45PPS

Chapter 6 Solutions

Glencoe Algebra 2 Student Edition C2014

Ch. 6.1 - Prob. 1AGP Ch. 6.1 - Prob. 1BGP Ch. 6.1 - Prob. 2AGP Ch. 6.1 - Prob. 2BGP Ch. 6.1 - Prob. 3AGP Ch. 6.1 - Prob. 3BGP Ch. 6.1 - Prob. 4GP Ch. 6.1 - Prob. 1CYU Ch. 6.1 - Prob. 2CYU Ch. 6.1 - Prob. 3CYU

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