Problem Solving

If the parabola represented by f(x)=ax^2+bx+c passes through points (-3,0), (0,3) and (5,0), which of the following must be true?

I. F(-1)>F(-2)

II. f(1)>f(0)

III. f(2) > f(1)


A. Only I
B. Only II
c. Only III
D. I and II
E. II and III

We are given three points on the parabola.

Plot those out on a quick graph so you get a quick mental picture of what is going on. It is a parabola with X-intercepts at (-3,0) and (5,0). They were nice to give these to us as points.

Parabolas have a point called the vertex. It is either the highest or lowest point on the curve depending on whether it opens upwards or downwards. It is also the axis of symmetry on the curve. The left and right sides of the curve are mirror images to one another based on this point.

When dealing with parabols the first thing you want to find is the vertex. Well, it can be tedious when you are given points and not the equation. BUT, we know that the vertex must lie centrally between the two x-intercepts. **NOTE that not all parabolas have an x-intercept.**

We can find the x component of the vertex by finding the point halfway between -3 and 5. This is at 1. This is the highest point on the parabola(f(x) is the greatest).

So, answering the questions now....

I. True. The curve is increasing as it moves from the left x-intercept towards the vertex.

II. True. f(1) is the greatest value on the curve.

III. False. f(1) is the greatest value.

Answer is D.