Question

What is the equation of the quadratic graph with a focus of (3, -1) and a directrix of y = 1?

Answer 1

Answer:

The equation of the quadratic graph is y =  $-\frac{1}{4}$ (x - 3)²

Step-by-step explanation:

The standard form of the equation of the quadratic graph is

(x - h)² = 4p(y - k), where

• The vertex of the parabola is (h, k)

• The focus is (h, k + p)

• The directrix is at y = k - p  

∵ The focus is (3, -1)

∵ The focus is (h, k + p)

→ Compare them

∴ h = 3

∴ k + p = -1 ⇒ (1)

∵ The directrix is at y = 1

∵ The directrix is at y = k - p

→ Compare them

∴ k - p = 1 ⇒ (2)

→ Add equations (1) and (2) to eliminate p

∵ (k + k) + (p - p) = (-1 + 1)

∴ 2k + 0 = 0

∴ 2k = 0

→ Divide both sides by 2

∴ K = 0

→ Substitute the value of k in equation (1) to find p

∵ 0 + p = -1

∴ p = -1

→ Substitute the values of h, k, and p in the form of the equation above

∵ (x - 3)² = 4(-1)(y - 0)

∴ (x - 3)² = -4(y)

∴ (x - 3)² = -4y

→ Divide both sides by -4

∴ $-\frac{1}{4}$ (x - 3)² = y

→ Switch the two sides

∴ y =  $-\frac{1}{4}$ (x - 3)²

∴ The equation of the quadratic graph is y =  $-\frac{1}{4}$ (x - 3)²