Question

A parabola has its maximum at (4, 8) and an x–intercept at x = –2.

Answer 1

The other x-intercept is at x = 10

Step-by-step explanation:

The vertex form of the parabola is y = a(x - h)² + k, where

• (h , k) are the coordinates of the vertex point of the parabola
• a is the leading coefficient of the equation of the parabola
• If a < 0 , then the vertex point is maximum
• If a > 0 , the the vertex point is minimum

∵ The parabola has a maximum point at (4 , 8)

∴ h = 4 and k = 8

∵ The form of the equation of the parabola is y = a(x - h)² + k

∴ The equation of the parabola is y = a(x - 4)² + 8

∵ The x-intercept is at x = -2

- x-intercept means the parabola intersects the x-axis at point (-2 , 0)

∵ x = -2 and y = 0

- Substitute these values in the equation above to find a

∴ 0 = a(-2 - 4)² + 8

∴ 0 = a(-6)² + 8

∴ 0 = 36 a + 8

- Subtract 36 a from both sides

∴ -36 a = 8

- Divide both sides by -36

∴ a = $\frac{-2}{9}$

∴ The equation of the parabola is y = $\frac{-2}{9}$ (x - 4)² + 8

To find the other x-intercept substitute y by 0

∵ 0 = $\frac{-2}{9}$ (x - 4)² + 8

- Subtract 8 from both sides

∴ -8 = $\frac{-2}{9}$ (x - 4)²

- Divide both sides by $\frac{-2}{9}$

∴ 36 = (x - 4)²

- Take √ for both sides

∴ x - 4 = 6 or x - 4 = -6

- Add 4 to both sides

∴ x = 10 or x = -2

The other x-intercept is at x = 10

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