find a quadratic function to model the values in the table

Question

x|y

-1|-8

0|-2

3|-8

Answer 1

The general form of a quadratic function is --> y = ax² + bx + c

Since (0, -2) exists for the function, we can plug in those values:

-2 = a(0)² + b(0) + c

-2 = 0 + 0 + c

-2 = c

So now our function so far is --> y = ax² + bx - 2

We have two pairs of coordinates left: (-1, -8) and (3, -8).

First, plug in the first pair and simplify as much as you can:

-8 = a(-1)² + b(-1) - 2

-8 = a - b - 2 ( add 2 to both sides )

-6 = a - b (stop here because we can't go further)

Second, plug in the second pair and simplify as much as you can:

-8 = a(3)² + b(3) - 2

-8 = 9a + 3b - 2 ( add 2 to both sides )

-6 = 9a + 3b (stop here because we can't go further)

Now we have these two equations left:

1. a - b = -6
2. 9a + 3b = -6

Now we solve for a and b using systems of equations, using one of three ways:

1. substitution
2. elimination
3. graphing (not my favorite, but it is doable)

Using substitution:

a - b = -6 can be rewritten as a = b - 6

plug into the second equation and solve for b

9(b - 6) + 3b = -6 (distribute the 9)

9b - 54 + 3b = -6 (combine all of the b's)

12b - 54 = -6 (add 54 to both sides)

12b = 48 (divide by 12 on both sides to isolate b)

b = 4

plug b into one of the original two equations

a - 4 = -6 (add 4 to both sides)

a = -2

The quadratic equation for this table is y = -2x² + 4b - 2