Question

Write an equation for each parabola.

1. A parabola with x-intercept at (-1, 0) and (3, 0) at which passes through the point (1, -8).

2. A parabola with x-intercept at 0 and 1 and at which passes through the point (2, -2).

Answer 1

Problem 1

Answer: y = 2x^2-4x-6

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Step-by-step explanation:

The x intercepts are -1 and 3, meaning that (x+1) and (x-3) are factors.

This is because x = -1 leads to x+1 = 0 when you add 1 to both sides. And x = 3 leads to x-3 = 0 when you subtract 3 from both sides.

The quadratic equation would be y = a(x+1)(x-3) = a(x^2-2x-3) for some constant 'a'. Use the point (x,y) = (1,-8) to find the value of 'a'.

y = a(x^2-2x-3)

-8 = a(1^2-2*1-3)

-8 = a(-4)

-4a = -8

a = -8/(-4)

a = 2

Therefore,

y = a(x^2-2x-3)

y = 2(x^2-2x-3)

y = 2x^2-4x-6

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Problem 2

Answer: y = -x^2+x

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Step-by-step explanation:

We'll follow the same idea as the previous problem.

The factors are (x-0) and (x-1). This is the same as saying the factors are x and (x-1)

So we have y = ax(x-1) = a(x^2-x)

Plug in (x,y) = (2,-2) and solve for 'a'.

y = a(x^2-x)

-2 = a(2^2-2)

-2 = a(2)

2a = -2

a = -2/2

a = -1

The equation updates to

y = a(x^2-x)

y = -1(x^2-x)

y = -x^2+x

Answer 2

Answer: 1) y = 2(x + 1)(x - 3)

2) y = -(x)(x - 1)

Step-by-step explanation:

Use the Intercept form of a quadratic equation: y = a(x - p)(x - q) where

• p and q are the x-intercepts (aka zeros)
• "a" is the vertical stretch
• (x, y) is another point on the line --> use to find the a-value

1) p = -1, q = 3, (x, y) = (1, -8)

y = a(x + 1)(x - 3)

-8 = a(1 + 1)(1 - 3)

-8 = a(2)(-2)

-8 = -4a

2 = a

Equation: y = 2(x + 1)(x - 3)

2) p = 0, q = 1, (x, y) = (2, -2)

y = a(x - 0)(x - 1)

-2 = a(2 - 0)(2 - 1)

-2 = a(2)(1)

-2 = 2a

-1 = a

Equation: y = -(x)(x - 1)