Given the points below, write the quadratic function that contains all 3.

Question

(-2, -6), (0, 6), (2, 2)

(2, 6), (1, 1), (-2, -2)

(-2, 7), (-1, 3), (3, 7)

(1, 0), (2, 4), (0, 2)



(2, -4), (3, -7), (1, -3)

Answer 1

Just for fun, since it's already been solved algebraically, solve with linear algebra

For 3 points

[x₁² x₁ 1][a] [y₁]

[x₂² x₂ 1][b] = [y₂]

[x₃² x₃ 1][c] [y₃]

So, for (-2,-6)(2,2)(0,6) the augmented matrix is

[4 -2 1 -6]

[4 2 1 2]

[0 0 1 6]

After Gaussian Elimination

[1 0 0 -2]

[0 1 0 2]

[0 0 1 6]

Which translates to a =-2, b = 2 and c = 6

Let's pick another one:

(1, 0), (2, 4), (0, 2)

[1 1 1 0]

[4 2 1 4]

[0 0 1 2]

After elimination

[1 0 0 3]

[0 1 0 -5]

[0 0 1 2]

a = 3, b = -5, c = 2

Arguably, this may not be easier, I think it is good to know.

However, you can use on line tools to do the Gaussian elimination for you once you have the matrix.

Answer 2

Any quadratic function can be written in the form y = ax² + bx + c

For any of these sets of 3 points, we can substitute the x and y values into the equation above. For the first set of 3 points, (-2, -6), (0, 6), (2, 2), we get

-6 = a(-2)² + b(-2) + c

6 = a(0)² + b(0) + c

2 = a(2)² + b(2) + c

These simplify to

-6 = 4a - 2b + c

6 = 0 + 0 + c

2 = 4a + 2b + c

or

-6 = 4a - 2b + c

6 = c

2 = 4a + 2b + c

We can start using systems of equations to solve for each variable (a, b, and c) to figure out the quadratic function.

For example, we can take the 1st and 3rd equations, and choose to subtract the 3rd equation from the 1st one.

-6 = 4a - 2b + c

- 2 = 4a + 2b + c

-8 = 0 - 4b + 0

-8 = -4b

/-4 /-4

2 = b

Equation 2 already tells us that c = 6

We can now choose an equation, put back in values of b = 2 and c = 6, and solve for a.

I have chosen equation 1:

-6 = a(-2)² + (2)(-2) + 6

-6 = 4a - 4 + 6

-6 = 4a + 2

-2 -2

-8 = 4a

/4 /4

-2 = a

Now that we have values for a, b, and c, we can now put them back into the general equation for a quadratic function: y = ax² + bx + c

y = -2x² + 2x + 6

To solve for the other quadratic functions, you can follow the same general method.

Answer 3

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