The graph of the function $y=a x^{2}+b x+c$ is shown in the diagram. Which of the following must be positive?
(A) $a$
(B) $b c$
(C) $a b^{2}$
(D) $b-c$
(E) $c-a$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-100.jpg?height=352&width=420&top_left_y=1296&top_left_x=1292)

## Part C: Each correct answer is worth 8.
Solution 1

A specific parabola that has this shape (that is, opening downwards and with a negative $x$ intercept that is more negative than the positive $x$-intercept is positive) is

$y=-(x+2)(x-1)=-x^{2}-x+2$.

In this parabola, $a=-1, b=-1, c=2$.

Using these values for $a, b$ and $c$, only $c-a$ is positive.

Since we are looking for the possibility which is positive no matter which parabola is used, then $c-a$ must be the answer.

## Solution 2

Since the parabola in the diagram opens downwards, then $a0$.

This tells us that $c-a$ is positive (since $a$ is negative) so must be the correct answer, since there is only one correct answer.

(We could check that $b$ is negative using the fact that the $x$-coordinate of the vertex is negative and so both $b c$ and $b-c$ must be negative.)

ANSWER: (E)