Challenge problem-Quadratic

[Guest]

Write an equation in the form y=ax^2 + bc +c for the quadratic function whose graph passes through (8,0) , (0,8) and ( -2,0)

I have no idea how to do this..I tohught that (-2.0) was the vertex..but I'm not sure.
The answer is confusing to understand for me: -1/2x^2 + 3x +8 I thought the quadraic would open up..cuz the postive 8s but it turns out it opens down?? HOW?

 

[Guest]

I thought the quadraic would open up..cuz the postive 8s but it turns out it opens down?? HOW?

When a quadratic opens down, that doesn't mean it is never above the x-axis. That means it is above the axis only between the two zeroes, if it has two zeroes. This one does: (8, 0) and (-2, 0). So it is above the x-axis between -2 and 8, and it is below the x-axis everywhere else. So we would expect that at 0, it is above the x-axis.

When you know the zeroes are r₁ and r₂, you can conclude that the equation is:

. . . . y = a(x - r₁)(x - r₂)

So plug in your r₁ and r₂. Then you can plug in the point (0, 8) to solve for a, and put a back into the equation with x and y.

 

[Gene]

Another way.
You have three equations in three unknowns.
(8,0) gives
0 = a*8^2+b8+c
(0,8) gives
8 = c and
( -2,0) gives
0 = a*(-2)^2+b*(-2)+c
We know c now so
0 = a*8^2+b8+8 = 64a+8b+8 = 8a+b+1
0 = a*(-2)^2+b*(-2)+8 = 4a-2b+8 = 8a-4b+16
Subtract
0 = 5b-15
b=3
We know b now so
0 = 4a-2*3+8
4a= -2
a=-1/2

The up or down key is a. Minus says it opens down.
The key to the vertex is (8,0) and ( -2,0). They are both x intercepts so The vertex is between them at x=(8+(-2))/2 = 3.
Now that you have the equation you can check that with
vertex is at x = -b/(2a) = -3/(2*(-1/2)) = 3

 

[soroban]

Hello, anna!

Write an equation in the form y = ax² + bc +c for the quadratic function
whose graph passes through (8,0) , (0,8) and (-2,0)

I thought that (-2.0) was the vertex..but I'm not sure. . . . Of course, you didn't make a sketch!

The answer is confusing to understand for me: -1/2x^2 + 3x +8
I thought the quadraic would open up..cuz the postive 8s . . . no!
but it turns out it opens down?? HOW? . . . Make a sketch!

             *(0,8)
             |                        Plot the three points.
             |                      (That wasn't hard, was it?)
             |
             |                       Did you REALLY think that
             |                         the vertex is (-2,0)?
   - - * - - + - - - - * - -           and it opens upward?
    (-2,0)           (8,0)

Gene had the best idea . . . "plug" the points into the equation.

The point (8,0) tell us: .x = 8 and y = 0
. . . We have: . 0 .= .a·8² + b·8 + c . . ---> . . 64a + 8b + c .= .0

The point (0,8) tell us: .x = 0 and y = 0
. . . We have: . 8 .= .a·0² + b·0 + c . . ---> . . c .= .8

The point (-2,0) tell us: .x = -2, y = 0
. . . We have: . 0 .= .a(-2)² + b(-2) + c . . ---> . . 4a - 2b + c .= .0

Now solve the system of equations . . .