Holt Algebra 2

10-5 Parabolas

Write the standard equation of a parabola and its axis of symmetry.

Graph a parabola and identify its focus, directrix, and axis of symmetry.

Objectives

directrix and focus of a parabola

Vocabulary

Holt Algebra 2

10-5 Parabolas

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.

Holt Algebra 2

10-5 Parabolas

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.

Holt Algebra 2

10-5 Parabolas

Holt Algebra 2

10-5 Parabolas

Notes

1. Write an equation for the parabola with vertex V(0, 0) and directrix y = 1.

4. Find the vertex, value of p, axis of symmetry, focus,

and directrix of the parabola y – 2 = (x – 4)2, 112

then graph.

3. Write an equation for the parabola with focus F(2, -3) and directrix x = 1.

2. Write an equation for the parabola with focus F(0, 0) and directrix y = -1.

Holt Algebra 2

10-5 Parabolas

Write the equation in standard form for the parabola.

Example 1A: Writing Equations of Parabolas

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. 14p

Step 2: Because p = 5 and the parabola opens

downward. The equation of the parabola is

y = – x2 1

20

Holt Algebra 2

10-5 Parabolas

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 124

Holt Algebra 2

10-5 Parabolas

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

Example 2A: Write the Equation of a Parabola

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

Holt Algebra 2

10-5 Parabolas

Write the equation of a parabola with focus

F(0, 4) and directrix y = –4.

Example 2B

Holt Algebra 2

10-5 Parabolas

Write the equation in standard form for each parabola.

vertex (0, 0), focus (0, –7)

Example 2C

The equation of the parabola is

Holt Algebra 2

10-5 Parabolas

Example 3: Graphing Parabolas

Step 1 The vertex is (2, –3).

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 2 , so 4p = 8 and p = 2. 1 4p

1 8

=

Holt Algebra 2

10-5 Parabolas

Example 3 Continued

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 3 The graph has a vertical axis of symmetry, with equation x = 2, and opens upward.

Holt Algebra 2

10-5 Parabolas

Step 1: The vertex is (1, 3) and p = 3

Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola. Then graph.

Example 4

Step 2 The graph opens right, with horizontal axis of symm. y = 3.

Step 4 The focus is (1 + 3, 3), or (4, 3).

Step 3 The directrix is vertical line x = –2.

Holt Algebra 2

10-5 Parabolas

Step 1: The parabola opens downward, the vertex is (8, 4) with axis of symm: x = 8.

Find the vertex, value of p axis of symmetry, focus, and directrix of the parabola. Then graph.

Example 5

Step 2: p = ½

Step 3: Use p to find F (8, 3.5)

and directrix: y = 4.5

Holt Algebra 2

10-5 Parabolas

Notes

1. Write an equation for the parabola with vertex V(0, 0) and directrix y = 1.

4. Find the vertex, value of p, axis of symmetry, focus,

and directrix of the parabola y – 2 = (x – 4)2, 112

then graph.

3. Write an equation for the parabola with focus F(2, -3) and directrix x = 1.

2. Write an equation for the parabola with focus F(0, 0) and directrix y = -1.