Question

Find an equation of a parabola with a vertex at the origin and directrix y = –2.5

Answer 1

Answer:

The equation of the parabola is:

$y=\dfrac{1}{10}x^2$

Step-by-step explanation:

Since the directrix is the line y = -2.5, this is a horizontal line ; thus the parabola opens up or down.

$(x - h)^2 = 4p (y - k)$ , where (h, k) is the vertex, the focus is (h, k + p), and the directrix is y = k - p.

The vertex is (h, k), or (0,0)

The directrix is y = -2.5

so k - p = -2.5

⇒    0 - p = -2.5

⇒ p = 2.5

The focus is (h, k + p)

⇒ (0, 0 +2.5)=(0, 2.5)

Thus substituting the values into the base equation,

$(x - h)^2 =4p(y-k)\\\\(x - 0)^2 = 4\times (2.5)\times (y-0)\\\\x^2 =10y\\\\y=\dfrac{1}{10}x^2$

Hence the equation of the parabola is:

$y=\dfrac{1}{10}x^2$

Answer 2

When the vertex is at the origin and the directrix is at y = -2.5, the the parabola must be facing upwards and have a general formula of y = 4a x^2. In this case, a is equal to 2.5 that is, y = 4*2.5 x^2 equal to y = 10 x^2