How to find the vertex of a graphed quadratic function

Question

I am wondering how to find the vertex/solve a quadratic function and turn it into standard form. The graph I have has the numbers (x,y) which are, (-9,0),(-7,8),(-5,8),and (-3,0). I am just confused because this graph does not show a vertex.

Answer 1

The intercept form of a quadratic equation i: y=a(x-p)(x-q) for a quadratic containing the points (p,0) and (q,0)

Your quadratic has the points (-9,0) and (-3,0) so its intercept form is y=a(x+9)(x+3)

Now we have to solve for a. We can do this by plugging in the point (-7,8):

8 = a(-7+9)(-7+3) = a(2)(-4) = -8a

Divide by -8 to solve for a

-1 = a

So the intercept form of the equation for your quadratic is y=-(x+9)(x+3)

Check your work by using the point (-5,8):

8 = -(-5+9)(-5+3) = -(4)(-2) = 8

Multiply out the intercept equation to find the standard equation:

y = -(x+9)(x+3)

y = -(x²+9x+3x+27) = -(x²+12x+27) = -x²-12x-27

The standard form of the equation for the quadratic is y=-x²-12x-27

You can graph either the intercept form or the standard form to find the vertex (they produce the same line). Alternatively, you can solve for the vertex form of the equation: y=a(x-h)²+k where (h,k) is the vertex.

We already know a=-1, so y = -(x-h)²+k

Let's plug in the point (-3,0):

0 = -(-3-h)²+k

-k -k

-k = -(-3-h)²

Let's plug in the point (-5,8)

8 = -(-5-h)²+k

-k-8 -k-8

-k = -(-5-h)²-8

(note: you could not use (-3,0) and (-9,0) or (-5,8) and (-7,8) as the pair of points you plug in because there are many quadratics that go through one of these pairs of points and have a=-1.)

Because these are both equal to -k, they are equal to each other:

-(-3-h)²=-(-5-h)²-8

divide by -1

(-3-h)²=(-5-h)²+8

multiply out the ²

9+6h+h² = 25-10h+h²+8 = 33+10h+h²

-9-6h-h² -9-6h-h²

0 = 24+4h

-4h -4h

-4h = 24

divide by -4

h=-6

Now plug h=-6 into the vertex form to solve for k: y=-(x+6)²+k

Let's use the point (-5,8) again:

8 = -(-5+6)²+k = -(1)²+k = -1+k

+1 +1

9 = k

Plug k=9 into the vertex form: y=-(x+6)²+9

(h,k) is the vertex, so the vertex is (-6,9)