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

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

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Mathematics
6 months
2021-07-29T20:33:44+00:00
2021-07-29T20:33:44+00:00 1 Answers
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## Answers ( )

Answer:The equation of the quadratic graph is

y =(x – 3)²Step-by-step explanation:The standard form of the equation of the quadratic graph is

, where(x – h)² = 4p(y – k)

∵ 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

∴ (x – 3)² = y

→ Switch the two sides

∴

y =(x – 3)²∴

The equation of the quadratic graph is y =(x – 3)²