9 1 Parabolas Parabolas Parabola the set of

9. 1 Parabolas

Parabolas Parabola: the set of points in a plane that are the same distance from a given point called the focus and a given line called the directrix. Notice that the axis of symmetry is the line through the focus, perpendicular to the directrix

d 2 d 1 Focus d 1 Vertex d 2 d 3 Directrix Notice that the vertex is located at the midpoint between the focus and the directrix. . . Also, notice that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix. . . Parabola with measures. gsp We can determine the coordinates of the focus, and the equation of the directrix, given the equation of the parabola. .

Standard Equation of a Parabola: (Vertex at the origin) Equation Focus Directrix 2 x = 4 py (0, p) y = –p (If the x-term is squared, the parabola is up or down) Equation y 2 = 4 px Focus (p, 0) Directrix x = –p (If the y-term is squared, the parabola is left or right) Where p is the direct distance from the vertex to the focus. * note: if p < 0, then parabola opens down or left

Standard Equation of a Parabola: (Vertex at (h, k) ) Equation 2 (x – h) = 4 p(y - k) Focus (h, k + p) Directrix y=k–p (If the x term is squared, the parabola is up or down) Equation 2 (y - k) = 4 p(x – h) Focus Directrix (h + p, k) x=h–p (If the y term is squared, the parabola is left or right)

Ex 1: Determine the focus and directrix of the parabola 2 y = 4 x : Since x is squared, the parabola goes up or down… 2 2 Solve for x x = 4 py 2 y = 4 x 4 4 x 2 = 1/4 y Solve for p 4 p = 1/4 p = 1/16 Focus: (0, p) Directrix: y = –p Focus: (0, 1/16) Directrix: y = – 1/16 Let’s see what this parabola looks like. . .

Ex 2: Determine the focus and directrix of the parabola 2 – 3 y – 12 x = 0 : Since y is squared, the parabola goes left or right… 2 2 Solve for y y = 4 px 2 – 3 y = 12 x – 3 y 2 = – 4 x Solve for p 4 p = – 4 p = – 1 Focus: (p, 0) Directrix: x = –p Focus: (– 1, 0) Directrix: x = 1 Let’s see what this parabola looks like. . .

Ex 3: Write the standard form of the equation of the parabola with focus at (0, 3) and vertex at the origin. Since the focus is on the y axis, (and vertex at the origin) the parabola goes up or down… x 2 = 4 py Since p = 3, the standard form of the equation is x 2 = 12 y Ex 4: Write the standard form of the equation of the parabola with directrix x = – 1 and vertex at the origin. Since the directrix is parallel to the y axis, (and vertex at the origin) the parabola goes left or right… y 2 = 4 px Since p = 1, the standard form of the equation is y 2 = 4 x

Summary… • • Opens Up / Down (x – h)2 = 4 p(y - k) p pos up p neg down Vertex: V (h , k) Axis: x = h Focus: F(h , k + p) Directrix: y = k – p 9_1 Parabola Examples • Opens Right / Left • (y - k)2 = 4 p(x – h) • • • p pos right p neg left Vertex: V (h, k) Axis: y = k Focus: F (h + p, k) Directrix: x = h – p

Try It Out • Given the equations below, – What is the focus? – What is the directrix?