Final answer:
The transformation of f(x) to h(x) entails a horizontal shift to the left by 5 units and a downward vertical shift by 2 units. For the given function f(x) = 3x² - 2, h(x) is calculated as h(x) = 3(x + 5)² - 4. Evaluating h(-3) gives us the value 8.
Step-by-step explanation:
To describe the transformation from f(x) to h(x) where h(x) = f(x + 5) - 2, we need to discern the effects of each alteration to the function.
The f(x + 5) inside the function h(x) translates to a horizontal shift of the graph of f(x) by 5 units to the left. This is because we are effectively taking the input value and reducing it by 5 before applying the function, which moves points on the graph to the left.
The - 2 outside the function indicates a vertical translation downward by 2 units. Each point on the graph of f(x) is moved 2 units in the negative y-direction.
Given f(x) = 3x² - 2, to find h(x), we substitute x with (x + 5) and then subtract 2:
h(x) = f(x + 5) - 2
= 3(x + 5)² - 2 - 2
= 3(x + 5)² - 4.
Now, to evaluate h(-3), we simply plug -3 into the h(x) formula:
h(-3) = 3(-3 + 5)² - 4
= 3(2)² - 4
= 3(4) - 4
= 12 - 4
= 8.
Thus, the value of h(-3) is 8.