How to find y of vertex of Parabola with y-intercept and x-intercepts

Question

I already know how to find the x of the vertex with this information but I do not know how to find the y of the vertex. How can I find the y? The x intercepts are (-1,0) (5,0) and the y intercept is (0,5) and the x of the vertex is 2.

Answer 1

From the roots ($x$ intercepts), you can write the equation of the parabola as: $y = a(x+1)(x-5)$

When $x = 0$ (this is the $y$ intercept), $a(-5) = 5$, so $a = -1$.

So $y = -(x+1)(x-5)$

Set $x=2$ for the vertex (turning point) to find $y = 9$.

(From what you wrote, I assume you know how to find the $x$ coordinate of the vertex or turning point, but just for completeness, it is the arithmetic mean (average) of the two real roots, so here it is simply $\frac{-1+5}2 = 2$).

Answer 2

Since $x=-1$ and $x=5$ are zeros of your parabola, your parabola is $$y=a(x+1)(x-5)$$ Since your y-intercept is $(0,5)$ you get $$5=a(1)(-5)$$ which implies $$a=-1$$

Thus your equation is $$y=-(x+1)(x-5)$$

You may write it in standard form $$y=-x^2+4x+5$$