Question

Given the Vertex (1,0) and the Directrix y = -¼, find the equation of the quadratic

(x - 1)2

(x + 5)2

2(x - 1)2

-x2 - 1

Answer 1

y = (x - 1)²

Explanation:

A parabola is the locus of a point such that its distance from a fixed point (known as the focus) and a fixed line (known as the directrix) is always constant. The equation of a parabola is a quadratic equation.

The equation of a parabola with a horizontal directrix is given by:

(x - h)² = 4p(y - k).

Where the directrix (y) = k - p, vertex = (h, k)

Given that: Vertex (1,0) and the Directrix y = -¼, Therefore:

vertex = (h, k) = (1, 0). h = 1 and k = 0.

y = k - p

but y = -1/4. Hence:

-1/4 = k - p

p = k + 1/4

p = 0 + 1/4

p = 1/4

Since the equation of the parabola is:

(x - h)² = 4p(y - k).

Substituting:

(x - 1)² = 4(1/4)(y - 0)

(x - 1)² = y

y = (x - 1)²

y = x² - 2x + 1