Question

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

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

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.