Question

Image: The parabola declines through 4, -1 drops to 5, -3 then increases to 6, 1.

What is the equation for the function in the graph?

Answer 1

I need one quick clarification: is the rightmost point (6, −1) or (6, 1)? A vertical parabola through (4, −1) and (6, −1) with vertex (5, −3) is consistent; (6, 1) is not symmetric about x = 5 and can't be the same vertical parabola.

If the points are (4, −1), (5, −3), (6, −1), the vertex is (5, −3) so
y = a(x − 5)^2 − 3.
Use (4, −1): −1 = a(1) − 3 → a = 2.
So the equation is y = 2(x − 5)^2 − 3 (or expanded: y = 2x^2 − 20x + 47).

If your rightmost point is actually (6, 1), please confirm or upload the image and I’ll re-evaluate.