Focus and DirectrixFocus and Directrix5-45-4

English Casbarro English Casbarro Unit 5: PolynomialsUnit 5: Polynomials

The standard form of a parabola:There are 2 situations:

1) Vertex at the origin: Focus is (0, p) and directrix is (0, –p ) x2 = 4py

2) Vertex at (h, k): Focus is (h, k + p) and directrix is (h, k – p)

Example 1

Find the coordinates of the focus and the equation for the directrixFor the parabola y = ¼x2.

First, let’s put it into the new form: x2 = 4py.

Since y = ¼x2 , you can multiply both sides by 4 for the new form: x2 = 4y . By substitution,

4y = 4py, so 4 = 4p. Thus, p = 1.

The coordinates of the focus: (0, k + 1) (0, 0 + p) (0, 0 + 1) (0, 1)The equation of the directrix: y = k – p y = 0 – p y = 0 – 1 y = –1

Example 2

The focus of a parabola has the coordinates (0, ) and the vertex is the origin . Find the equation of the parabola.

The coordinates of the focus : (0, ) which means that k + p = , and since k = 0, then the equation is 0 + p = . So, p = .

The new form we’re using (the vertex is the origin): x 2 = 4py The equation of this parabola, then is x2 = 4( ) y, so x2 = (-20/2)y.

Then equation is: x2 = -10y

Example 3

The directrix of a parabola is y = –2 and the focus is (0, 2). Find the equation of the parabola.

Since the focus is on the y-axis, that means that the vertex is the origin.

The coordinates of the focus : (0, 2 ) which means that k + p = 2, and since k = 0, the equation is 0 + p = 2 . So, p = 2.

The new form we’re using (the vertex is the origin): x 2 = 4py The equation of this parabola, then is x2 = 4(2) y, so x2 = 8y.Then equation is: x2 = 8y

Example 4

A parabola has vertex (–2, 4) and focus ( –2, ). Write the equation of the Directrix and the parabola.

We will have to use the new form that includes non-zero h and k: (x – h) 2 = 4p(y – k)

The coordinates of the focus : (–2, ) which means that k + p = , and since k = -2, the equation is -2 + p = .

So, p = - 2 p = . The equation of the directrix is: y = k – p y = - y = 2The equation of this parabola, then is (x +2) 2 = 4(2)(y – 4)

The equation is: (x + 2)2 = 8(y – 4)

Example 5

A parabola has vertex (2, –5), focus at (2, –3), and directrix y = –7. Write the equation in the different forms:

a) (x – h)2 = 4p(y – k) b) vertex form c) standard form