1. Focus and Directrix 5-4 English Casbarro Unit 5: Polynomials

2. The standard form of a parabola: There are 2 situations: 1)Vertex at the origin: Focus is (0, p) and directrix is (0, –p ) x 2 = 4py 2) Vertex at (h, k): Focus is (h, k + p) and directrix is (h, k – p)

3. Example 1 Find the coordinates of the focus and the equation for the directrix For the parabola y = ¼x 2. First, let’s put it into the new form: x 2 = 4py. Since y = ¼x 2, you can multiply both sides by 4 for the new form: x 2 = 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

4. 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 x 2 = 4( ) y, so x 2 = (-20/2)y. Then equation is: x 2 = -10y

5. 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 x 2 = 4(2) y, so x 2 = 8y. Then equation is: x 2 = 8y

6. 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 = 2 The equation of this parabola, then is (x +2) 2 = 4(2)(y – 4) The equation is: (x + 2) 2 = 8(y – 4)

7. 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