Write an equation for the quadratic function whose graph is shown below.

Question

It goes upwards. Points are:

(-9,0), (2,0), (3,2)

Answer 1

I cannot see the graph, but you have provided enough information for an answer.

We can substitute each pair into the equation y = ax² + bx + c and solve by elimination.

y = a (-9)² + b (-9) + c = 81a - 9 b + c = 0

y = a ( 2)² + b (2 ) + c = 4a + 2 b + c = 0

y = a (3 )² + b (3) + c = 9a + 3b + c = 2

To keep track of my steps, I like to name my equations. So I will call the first one Abe, the second one Ben, and the third one Cal. This are just my random identifications.

So I want to take 2 equations and subtract them to eliminate one variable. I choose to eliminate c.

Abe 81a - 9 b + c = 0

-Ben. - (4a + 2 b + c = 0)

=Don. 77 a - 11 b = 0. Now divide this equation by 11. 7a - b = 0 or b = 7a

Next I choose to eliminate the SAME variable c, from two different equations. I choose Abe and Cal.

Abe. 81a - 9 b + c = 0

-Cal -( 9a + 3b + c = 2)

=Ed 72a - 12b = -2

Now substitute the equation named Don into the equation named Ed to eliminate b.

Don into Ed 72a - 12(7a) = -2

Solve for a 72a - 84a = -2

- 12a = -2

a = -2/ -12

a = 1/6

Put this value for a into Don b = 7a

b = 7 (1/6)

b = 7/6

Put both of these values into Cal to get c.

Cal 9a + 3b + c = 2

9(1/6) + 3(7/6) + c = 2

1.5 + 3.5 + c = 2

5 + c = 2

c = --3

Answer: y = (1/6) x² + (7/6) x -3

Answer 2

Since (-9,0) and (2,0) are on the graph, -9 and 2 are roots of the quadratic. So, x - 2 and x- (-9) are factors.

So, y = a(x - 2)(x + 9)

Since (3,2) is on the graph, 2 = a(3-2)(3+9). Solve for a.