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 [tex]f(x)[/tex] about the x-axis can be done by changing [tex]f(x)[/tex] into [tex]-f(x)[/tex]. This means that if we have a line in the form
[tex]y=mx+q[/tex]
a reflection about the x-axis can be done by changing the function into
[tex]y=-mx-q[/tex]
2- A reflection of [tex]f(x)[/tex] about the y-axis can be done by changing [tex]f(x)[/tex] into [tex]f(-x)[/tex]. This means that if we have a line in the form
[tex]y=mx+q[/tex]
a reflection about the x-axis can be done by replacing all the x with -x:
[tex]y=-mx+q[/tex]
Back to our exercise:
The original function is [tex]f(x)=-3x-8[/tex]. The final function is [tex]g(x)=-3x+12[/tex]. 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
[tex]f'(x)=-3x+8[/tex]
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
[tex]f'(x)+4=-3x+8+4=-3x+12[/tex]
which corresponds to g(x). So, the correct option is D.
