What is the equation in the standard form of a parabola that contains the following points (-2, 18), (0, 2), (4, 42)?

Solution:

The equation of a parabola in a standard form is expressed as y = f(x) = ax² + bx + c where a, b, and c are constants and a ≠ 0.

Now we are given three coordinates (-2, 18), (0, 2) and (4, 42) which lie on the parabola and which means that it satisfies the parabolic equation.

To find the values of a, b and c we need to formulate three equations using the three given points. Since there are three unknowns a, b, and c, we will require three equations.

1. Point (-2, 18) i.e. x = -2 and y = 18

Substituting the values of x and y in the parabola equation y = ax² + bx + c we get

a(-2)² + b(-2) + c = 18

4a - 2b + c = 18 --- (1)

2. Point (0, 2) i.e. x = 0 and y =2

a(0)² + b(0) + c = 2

c = 2 --- (2)

3. Point (4, 42) i.e. x = 4, y = 42

a(4)² + b(4) + c = 42

16a + 4b + c = 42 --- (3)

Substituting equation (2) in (1) & (3) we get two equations in parameters a and b

4a - 2b + 2 = 18

4a - 2b = 16 --- (4)

16a + 4b + 2 = 42

16a + 4b = 40 --- (5)

Solving equations (4) and (5)

4a - 2b = 16 --- (i)

16a + 4b = 40 --- (ii)

Multiply (i) by 2 and add it with (ii)

8a - 4b = 32
16a + 4b = 40
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24a + 0 = 72
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⇒ 24a = 72

⇒ a = 3

Substituting a = 3 in (i) we get

⇒ 4 × 3 - 2b = 16

⇒ 12 - 2b = 16

2b = 12 - 16

⇒ 2b = -4

⇒ b = -2

Therefore the parabolic equation in standard form is: y = 3x² - 2x + 2

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What is the equation in the standard form of a parabola that contains the following points (-2, 18), (0, 2), (4, 42)?

Summary:

The equation in the standard form of a parabola that contains the following points (-2, 18), (0, 2), (4, 42) is y = 3x² - 2x + 2.