Question

Which series of transformations takes the graph of f(x)=3x−8 to the graph of g(x)=−3x+12 ?

A.) reflect the graph about the y-axis and translate 4 units down

B.) reflect the graph about the y-axis and translate 4 units up

C.) reflect the graph about the x-axis and translate 4 units down

D.) reflect the graph about the x-axis and translate 4 units up

Answer 1

reflect the graph about the x-axis and translate 4 units down

Answer 2

D.) reflect the graph about the x-axis and translate 4 units up

Step-by-step explanation:

Before to identify the correct choice, let's see the definition of reflection along the x-axis and the y-axis:

1- A reflection of
`f(x)` about the x-axis can be done by changing
`f(x)` into
`-f(x)`. This means that if we have a line in the form

`y=mx+q`

a reflection about the x-axis can be done by changing the function into

`y=-mx-q`

2- A reflection of
`f(x)` about the y-axis can be done by changing
`f(x)` into
`f(-x)`. This means that if we have a line in the form

`y=mx+q`

a reflection about the x-axis can be done by replacing all the x with -x:

`y=-mx+q`

Back to our exercise:

The original function is
`f(x)=-3x-8`. The final function is
`g(x)=-3x+12`. First of all, we immediately notice that both the signs of m and q have been changed: therefore, it must be a reflection about the x-axis, so we can discard option B.

The reflection of f(x) about the x-axis is

`f'(x)=-3x+8`

We see that the y-intercept is +8, while in g(x) the y-intercept is +12. In order to match the two functions, we must translate f'(x) up by 4 units, so that we get

`f'(x)+4=-3x+8+4=-3x+12`

which corresponds to g(x). So, the correct option is D.