6 minutes Warm-Up For each parabola, find an equation for the axis of symmetry and the coordinates of the vertex. State whether the parabola opens up or down, and whether the y-coordinate of the vertex is the minimum or maximum value of the function. 1) 2)

9.2 Parabolas Objectives: Write and graph the standard equation of a parabola given sufficient information Given an equation of a parabola, graph it and label the vertex, focus, and directrix

Parabolas A parabola is defined in terms of a fixed point, called the focus, A parabola is the set of all points P(x,y) in the plane whose distance to the focus focus and a fixed line, called the directrix. equals its distance to the directrix. directrix axis of symmetry

Horizontal Directrix Standard Equation of a parabola with its vertex at the origin is y = x2 1 4p x y D(x, –p) P(x, y) F(0, p) y = –p O p > 0: opens upward p < 0: opens downward focus: (0, p) directrix: y = –p axis of symmetry: y-axis

Vertical Directrix Standard Equation of a parabola with its vertex at the origin is x = y2 1 4p x y D(x, –p) P(x, y) F(p, 0) x = –p O p > 0: opens right p < 0: opens left focus: (p, 0) directrix: x = –p axis of symmetry: x-axis

Example 1 Graph . Label the vertex, focus, and directrix. Identify p. -4 -2 2 4 So, p = 1 Since p > 0, the parabola opens to the right. Vertex: (0,0) Focus: (1,0) Directrix: x = -1

Example 1 Graph . Label the vertex, focus, and directrix. y x Use a table to sketch a graph -4 -2 2 4 y x 2 4 -2 -4 1 4 1 4

Example 2 Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6. Since the directrix is below the vertex, the parabola opens up Since y = -p and y = -6, p = 6

Standard Equation of a Translated Parabola Horizontal Directrix: y – k = (x – h)2 1 4p vertex: (h, k) focus: (h, k + p) directrix: y = k – p axis of symmetry: x = h

Standard Equation of a Translated Parabola Vertical Directrix: x – h = (y - k)2 1 4p vertex: (h, k) focus: (h + p, k) directrix: x = h - p axis of symmetry: y = k

Example 3 Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. The parabola opens downward, so the equation is of the form y – k = (x – h)2 1 4p vertex: (-3,3) h = -3, k = 3 p = -1 y – 3 = (x + 3)2 1 -4

Practice Write the standard equation of the parabola with its focus at F(-6,4) and directrix x = 2.

Example 4 Graph the parabola . Label the vertex, focus, and directrix. -4( ) ( )-4 Isolate the y-terms Complete the square

Example 4 Graph the parabola . Label the vertex, focus, and directrix. x – h = (y - k)2 1 4p directrix: x = 0 vertex: (h, k) = (1,-2) Find p: so, p = 1 focus: F(2,-2) focus: = (2,-2) directrix: x = 0 vertex: V(1,-2)

Example 4 Graph the parabola . Label the vertex, focus, and directrix. directrix: x = 0 y x 2 -2 -4 -6 focus: F(2,-2) 2 5 1 2 5 vertex: V(1,-2)

Practice Graph the parabola x2 – 6x + 6y + 18 = 0. Label the vertex, focus, and directrix.

Homework worksheet