Objectives Write the standard equation of a parabola and its axis of symmetry. Graph a parabola and identify its focus, directrix, and axis of symmetry. Vocabulary directrix and focus of a parabola

A parabola is the set of all points P(x, y) in a plane that are an equal distance from both a fixed point, the focus, and a fixed line, the directrix. A parabola has a axis of symmetry perpendicular to its directrix and that passes through its vertex. The vertex of a parabola is the midpoint of the perpendicular segment connecting the focus and the directrix.

Previously, you have graphed parabolas with vertical axes of symmetry that open upward or downward. Parabolas may also have horizontal axes of symmetry and may open to the left or right. The equations of parabolas use the parameter p. The |p| gives the distance from the vertex to both the focus and the directrix.

Notes 1. Write an equation for the parabola with vertex V(0, 0) and directrix y = 1. 2. Write an equation for the parabola with focus F(0, 0) and directrix y = -1. 3. Write an equation for the parabola with focus F(2, -3) and directrix x = 1. 4. Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola y – 2 = (x – 4)2, 1 12 then graph.

Example 1A: Writing Equations of Parabolas Write the equation in standard form for the parabola. Step 1 Because the axis of symmetry is vertical and the parabola opens downward, the equation is in the form y = x2 with p = 5. 1 4p Step 2: Because p = 5 and the parabola opens downward. The equation of the parabola is y = – x2 1 20

Example 1B: Writing Equations of Parabolas Write the equation in standard form for the parabola with vertex (0, 0) and directrix x = -6 Step 1 Because the directrix is a vertical line, the equation is in the form and the graph opens to the right. Step 2 The equation of the parabola is . x = y2 1 24 7

Example 2A: Write the Equation of a Parabola Write the equation of a parabola with focus F(2, 4) and directrix y = –4. Step 1 The vertex is (2,0)… halfway between the focus and directrix. Step 2 The equation opens up so and p = 4. y = x2 1 4p Step 3

Example 2B Write the equation of a parabola with focus F(0, 4) and directrix y = –4.

Example 2C Write the equation in standard form for each parabola. vertex (0, 0), focus (0, –7) The equation of the parabola is

Example 3: Graphing Parabolas Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola Then graph. y+3 = (x – 2)2. 1 8 Step 1 The vertex is (2, –3). Step 2 , so 4p = 8 and p = 2. 1 4p 8 =

Example 3 Continued Step 3 The graph has a vertical axis of symmetry, with equation x = 2, and opens upward. Step 4 The focus is (2, –3 + 2), or (2, –1). Step 5 The directrix is a horizontal line y = –3 – 2, or y = –5.

Step 1: The vertex is (1, 3) and p = 3 Example 4 Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola. Then graph. Step 1: The vertex is (1, 3) and p = 3 Step 2 The graph opens right, with horizontal axis of symm. y = 3. Step 3 The directrix is vertical line x = –2. Step 4 The focus is (1 + 3, 3), or (4, 3).

Example 5 Find the vertex, value of p axis of symmetry, focus, and directrix of the parabola. Then graph. Step 1: The parabola opens downward, the vertex is (8, 4) with axis of symm: x = 8. Step 2: p = ½ Step 3: Use p to find F (8, 3.5) and directrix: y = 4.5

Notes 1. Write an equation for the parabola with vertex V(0, 0) and directrix y = 1. 2. Write an equation for the parabola with focus F(0, 0) and directrix y = -1. 3. Write an equation for the parabola with focus F(2, -3) and directrix x = 1. 4. Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola y – 2 = (x – 4)2, 1 12 then graph.