How do I calculate the equation for this unique parabola?

Question

Y-intercepts: (0, 8) and (0, -6) 
Vertex: (10, 0)

Answer 1

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

Substitute vertex (10,0) in the equation.

x = a(y - 0)² + 10

x = ay² + 10  -> (1)

Taking first y-intercept:

Substitute Y-intercept is (0,8) in equation (1).

0 = a(8)² + 10

0 = 64a + 10

64a = -10

a = -10/64 = 0.49.

Therefore the parabolic equation is x = -0.49y² + 10 (vertex form).

Taking second y-intercept:

Substitute Y-intercept is (0,-6) in standard equation (1).

0 = a(-6)² + 10

0 = 36a + 10

36a = -10

a = -10/36 = -0.27.

Therefore the parabolic equation is x = -0.27y² + 10 (vertex form).

Therefore the parabolic equations in vertex form are x = -0.49y² + 10 and x = -0.27y² + 10.

 

Answer 2

The y - intercepts of the parabola are (0, 8) and (0, - 6) and vertex is (10, 0).

Here the parabola contains two y - intercepts and the x - coordinate of the vertex is 10. So, the general form of parabola equation is x = ax² + bx + c, where a ≠ 0.

The intecept form of quadratic equation(parabola equation) is x = a(y - p)(y - q), where p and q are y - intercepts, and

• The role of ' a '.

1. If a > 0, then the parabola opens right.
2. If a < 0, then the parabola opens left.

The intecept form of quadratic equation (parabola equation) is x = a(y - 8)(y - (- 6))

⟹ x = a(y - 8)(y + 6)

⟹ x = a(y² - 2y - 48)

Vertex = (10, 0)

Substitute the value of (x, y) = (10, 0) in the above equation.

⟹ 10 = a[(0)² - 2(0) - 48]

⟹ 10 = - 48a

⟹ a = - 10/48 = - 5/24

 

⟹ x = (- 5/24)[y² - 2y - 48]

⟹ x = (- 5/24)y² + (5/12)y + 10.

However that make the vertex (10.2083, 1) so there is no parabola that fit all 3 of those points [(0, 8), (0, - 6) and (10, 0)].