In the diagram, the parabola has $x$-intercepts -1 and 4 , and $y$-intercept 8. If the parabola passes through the point $(3, w)$, what is the value of $w$ ?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-075.jpg?height=426&width=504&top_left_y=1462&top_left_x=1296)
Solution 1

A parabola is symmetric about its axis of symmetry.

Since the $x$-intercepts of the given parabola are $x=-1$ and $x=4$, then the axis of symmetry of the parabola is $x=\frac{-1+4}{2}=\frac{3}{2}$.

Since the point $(3, w)$ is $\frac{3}{2}$ units to the right of the axis of symmetry, then its $y$-coordinate (namely $w$ ) equals the $y$-coordinate of the point $\frac{3}{2}$ units to the left of the axis of symmetry, which is the point with $x=0$.

When $x=0$, we know that $y=8$.

Therefore, $w=8$.

(We could also note that $x=3$ is 1 unit to the left of the rightmost $x$-intercept so its $y$ coordinate is equal to that of the point 1 unit to the right of the leftmost $x$-intercept.)

## Solution 2

Since the parabola has $x$-intercepts of -1 and 4 , then its equation is of the form $y=a(x+1)(x-4)$ for some value of $a$.

Since the point $(0,8)$ lies on the parabola, then $8=a(1)(-4)$ or $a=-2$.

Therefore, the parabola has equation $y=-2(x+1)(x-4)$.

Since the point $(3, w)$ lies on the parabola, then $w=-2(4)(-1)=8$.

ANSWER: (E)