For each of the following functions f (x) and g(x),

banganX

banganX

Answered question

2021-08-16

For each of the following functions f (x) and g(x), express g(x) in the form a:f(x+b)+c for some values a,b and c, and hence describe a sequence of horizontal and vertical transformations which map f(x)  g(x).(a)(i)
f(x)=x2,g(x)=2x2+4x
(ii)f(x)=x2,g(x)=3x224x+8
(b)(i)f(x)=x2+3,g(x)=x26x+8
(ii)f(x)=x22,g(x)=2+8x4x2

Answer & Explanation

Daphne Broadhurst

Daphne Broadhurst

Skilled2021-08-17Added 109 answers

Step 1
a)(i) Given that: f(x)=x2, and g(x)=2x2+4x
g(x)=2x2+4x
Factor out common factor 2.
g(x)=2(x2+2x)
Complete the square for x2+2x
g(x)=2(x2+2x+1)2
Apply the rule: a2+2ab+b2=(a+b)2
g(x)=2(x+1)22=2f(x+1)2
The transformation is a horizontal translation (10) and a vertical stretch of a scale factor 2 followed by a vertical translation (02)
(ii) Given that: f(x)=x2, and g(x)=3x224x+8
g(x)=3x224x+8
Factor out common factor 3.
g(x)=3(x28x)+8
Complete the square for x28x
g(x)=3(x28x+16)40
Apply the rule: a2+2ab+b2=(a+b)2
g(x)=3(x4)240=3f(x4)40
The transformation is a horizontal translation (40) and a vertical stretch of a scale factor 3 followed by a vertical translation (040)
Step 2
b)(i)
Given that: f(x)=x2+3, and g(x)=x26x+8
g(x)=x26x+8
Complete the square for x26x+8.
g(x)=(x26x+9)1
Apply the rule: a2+2ab+b2=(a+b)2

Jeffrey Jordon

Jeffrey Jordon

Expert2021-10-10Added 2605 answers

Answer is given below (on video)

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