Question

Comprare the graph to the graph of f(x)=x^2
1. g(x) = 2(x + 5)^2
2. h(x) = −(x − 2)^2
3. g(x) = 3(x − 1)^2+6
4. h(x) = 1/2(x + 4)^2-2
All I need is the translation from these compared to f(x)=x^2

Answer 1

We'll transform from each of these so their image is x^2

1. g(x)=2(x+5)^2

Translate 5 to the right

x' = x + 5

Now we have 2x^2 which is twice as tall as x^2

Scale the y component by 1/2

y' = y / 2

So our transformation (x,y)->(x+5, y/2)  maps g(x) to f(x)

2. h(x) = -(x-2)^2

Translate left by 2,

x' = x - 2

It's upside down, reflect in the x axis

y' = -y

Transformation: (x,y)->(x - 2, -y) maps h(x) to f(x)

3. g(x) = 3(x- 1)^2 + 6

Translate left 1

x' = x - 1

Translate down 6

y' = y - 6

Scale y by 1/3

y'' = y'/3 = (y - 6)/3 = y/3 - 2

Transformation: (x,y)->(x - 1, y/3 - 2)

4. h(x) = (1/2) (x+4)^2 - 2

Translate right 4

x' = x + 4

Translate up 2

y' =  y + 2

Scale y by 2

y'' = 2y' = 2y + 4

Transformation: (x,y)->(x + 4, 2y + 4)