Question

Find a quadratic function to model the values in the table. Predict the value of y for x = 6.
x | y
-1 2
0 -2
3 10

Answer 1

are you familiar with polynomial interpolation?

Answer 2

no i am not.

Answer 3

y = ax^2 + bx + c

is your basic general quadratic equation

When x = -1 and y = 2, we plug them in to get

y = ax^2 + bx + c

2 = a(-2)^2 + b(-2) + c

2 = 4a - 2b + c

4a - 2b + c = 2

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When x = 0 and y = -2, we plug them in to get

y = ax^2 + bx + c

-2 = a(0)^2 + b(0) + c

-2 = 0a + 0b + c

-2 = 0 + 0 + c

-2 = c

c = -2

This means that 4a - 2b + c = 2 becomes 4a - 2b + (-2) = 2 which turns into 4a - 2b = 4

Answer 4

When x = 3 and y = 10, we plug them in to get

y = ax^2 + bx + c

10 = a(3)^2 + b(3) + c

10 = 9a + 3b + c

10 = 9a + 3b + (-2)

9a + 3b = 12

Answer 5

So we now have this system of equations

4a - 2b = 4
9a + 3b = 12

Answer 6

oh wait i made a typo I'm just catching now...

Answer 7

When x = -1 and y = 2, we plug them in to get

y = ax^2 + bx + c

2 = a(-1)^2 + b(-1) + c

2 = a - b + c

a - b + c = 2

a - b + (-2) = 2

a - b = 2 + 2

a - b = 4

Answer 8

So the system is

a-b = 4
9a + 3b = 12

Solve this to find 'a' and 'b'

Answer 9

a = 2 and b = -2?

Answer 10

good, so a = 2, b = -2 and c = -2

So the quadratic equation is

y = 2x^2 - 2x - 2

Answer 11

Your last step is to plug in x = 6 and evaluate to find y.

Answer 12

y=70-2x

Answer 13

y = 2x^2 - 2x - 2

y = 2(6)^2 - 2(6) - 2

y = 2(36) - 2(6) - 2

y = 72 - 12 - 2

y = 60 - 2

y = 58

Answer 14

So when x = 6, y is y = 58

Answer 15

oh okay I got it. Thanks for helping!!!! :)

Answer 16

np