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If the parabola represented by f(x) = ax^2 + bx + c

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If the parabola represented by f(x) = ax^2 + bx + c

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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. I and III

The OA is B.

Do I need to make the graph of the parabola? Is there another way to solve this PS question? Experts, I would like some help.

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On this problem, doing a quick sketch could be really useful to visualize the information you have, but it isn't strictly necessary.

We know that the parabola passes through both (-3,0) and (5,0). Since these points both have the same y-coordinate, we know that the vertex of the graph must have an x-coordinate that is directly between these two points. This gives us an x-coordinate of 1 for the vertex. We also know that our parabola can either face up or down, so all of the points between (-3,0) and (5,0) must either fall above the x-axis or below the x-axis.

Our third point tells us that (0,3) is on the graph. This point comes between (-3,0) and (5,0), and it falls above the x-axis. This means that our vertex must be above the x-axis, our vertex must be the highest point on our graph, and our parabola must be facing down.

Using that information, we can evaluate each of the statements.

I. f(-1) is farther from our vertex [f(1)] than f(2). This means it is farther down the parabola, making f(-1) < f(2). FALSE.
II. f(1) is the vertex on a downwards facing parabola, so it is the highest point on our graph. So f(1) > f(anything). TRUE.
III. Again, f(1) is the highest point on the graph, so nothing can be > f(1). FALSE.

This gives B as the correct answer! If you had trouble following any of this, let me know, and I can demonstrate with diagrams. Smile

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