Question

If f(x) = x^2 and g(x) = 1/2 f(x + 3) + 6 Describe the transformation(s)​

Answer 1

First, let's define some transformations:

Horizontal translation:

For a function f(x) a horizontal translation of N is written as

g(x) = f(x - N)

If N is positive we have a translation to the right

if N is negative we have a translation to the left.

Vertical translation:

For a function f(x) a vertical translation of N is written as

g(x) = f(x) + N

If N is positive we have a translation up

if N is negative we have a translation down

Vertical dilation.

For a function f(x), a vertical dilation of scale factor k is written as:

g(x) = k*f(x)

Now we can see that in our problem we have:

g(x) = (1/2)*f(x + 3) + 6

Then if we start wit f(x) we have:

first we have a dilation of scale factor k = 1/2

g(x) = (1/2)*f(x)

then we have a translation of 6 units up

g(x) = (1/2)*f(x) + 6

then a translation of 3 units to the left

g(x) = (1/2)*f(x - 3) + 6