Question

How do you use the important points to sketch the graph of `y=-x^2+12x-36`?

Answer 1

The important points for a graph sketch are the zeros/roots and inflection points.

Explanation:

This one is a quadratic, so it will have the shape of a parabola. With a negative `x^2` coefficient the inflection point will be a maximum.

f’(x) = -2x + 12 ; f”(x) = -2 Thus, it is a maximum at x = -2. (y = -64)
An easy point to calculate is x=0: (0, -36) Taking the same interval on the other side gives us a symmetric point to sketch the parabola. 0 is 2 units away from -2, so 2 units on the other side is x = -4.
Calculating this gives us the point: (-4, -100)

The zeros/roots can be calculated from the quadratic formula.