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.
