What is the equation in the standard form of a parabola that models the values in the table?

x	-2	0	4
f(x)	0	-6	78

Solution:

The standard equation of a parabola is

y = ax² + bx + c

Substitute the given values in this equation

When x = - 2 and y = 0

0 = a (-2)² + b (-2) + c

0 = 4a - 2b + c …. (1)

When x = 0 and y = - 6

-6 = a (0)² + b(0) + c

-6 = c …. (2)

When x = 4 and y = 78

78 = a (4)² + b (4) + c

78 = 16a + 4b + c ….. (3)

Equation (1) and (3) can be written as

4a - 2b + c = 0 …. (4)

16a + 4b + c = 78 …. (5)

Multiply equation (4) by 4 to eliminate 'a'

16a - 8b + 4c = 0

16a + 4b + c = 78

Subtracting these equations

-12b + 3c = - 78

Substitute the value of 'c' here

-12b + 3 (-6) = - 78

-12b - 18 = - 78

-12b = - 78 + 18

-12b = -60

Divide both sides by -12

b = 5

Substitute the value of b and c in equation (1)

16a + 4b + c = 78

16a + 4 (5) + (-6) = 78

16a + 20 - 6 = 78

By further calculation

16a = 78 - 20 + 6

16a = 64

Divide both sides by 16

a = 4

Here the value of a = 4, b = 5 and c = -6.

So the equation formed is y = 4x² + 5x - 6

Therefore, the equation of the parabola is y = 4x² + 5x - 6.

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What is the equation in the standard form of a parabola that models the values in the table?

x	-2	0	4
f(x)	0	-6	78

Summary:

The equation in the standard form of a parabola that models the values in the table is y = 4x² + 5x - 6.