The equation for a parabola has the form y = ax2 + bx + c, where a, b, and c are constants and a ≠ 0. find an equation for the parabola that passes through the points (−1, −10), (1, −6), and (2, −13).

To solve the problem, substitute the given points for x in the given equation to get

[tex]-10=(-1)^2a+(-1)b+c\Rightarrow-10=a-b+c \\ -6=(1)^2a+(1)b+c\Rightarrow-6=a+b+c \\ -13=(2)^2a+(2)b+c\Rightarrow-13=4a+2b+c[/tex]

Solving the three equations simultaneously, we have:

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

Therefore, the required equation is

[tex]y=-3x^2+2x-5[/tex]

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eudora eudora · Answer 2 · 2021-08-24T19:49:15+02:00

Equation of a parabola is given by,

[tex]y=ax^2+bx+c[/tex]

If the graph of the given parabola passes through three points (-1, -10), (1, -6) and (2, -13).

For (-1, -10),

-10 = a(-1)² + b(-1) + c

a - b + c = -10 -------(1)

For (1, -6),

-6 = a(1)² + b(1) + c

a + b + c = -6 --------(2)

For (2, -13),

-13 = a(2)² + b(2) + c

4a + 2b + c = -13 -------(3)

Add equation (1) and (2),

(a - b + c) + (a + b + c) = -10 + (-6)

2a + 2c = -16

a + c = -8 ---------(4)

Multiply equation (2) by 2 and subtract it from equation (3),

(4a + 2b + c) - 2(a + b + c) = -13 + 6(2)

2a - c = -1 --------(5)

Add equation (4) and equation (5),

(a + c) + (2a - c) = -8 - 1

3a = -9

a = -3

By substituting the value of 'a' in equation (4),

-3 + c = -8

c = -5

By substituting the value of a and c in equation (1),

-3 - b - 5 = -10

b + 8 = 10

b = 2

Therefore, equation of the parabola will be,

-3x² + 2x - 5 = 0

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