Warm-Up to Check Prior Knowledge

1.What is the midpoint formula?

2. What is the midpoint of (2, 3) and (2, – 7)?

3. What is the midpoint of (– 2, – 1) and (5, – 1)?

4. What is the distance between (2, – 2) and (2, – 7)?

5. What is the distance between (– 4, – 3) and (6, – 3)?

6. What value will complete the square of x² + 14x + ?

7. Rewrite y = 2x² – 12x + 24 in vertex form.

(2, – 2)

(1.5, – 1)

5 units

10 units

49

Parabolas and Their Equations

Learning Objectives:Given a focus and directrix you will be able to derive the equation of a vertical parabola y = a(x - h) 2 + k, and derive the equation of a horizontal parabola x = a(y - k) 2 + h, where (h, k) is the vertex of the parabola.

You will be able to solve real-world problems involving parabolas.

c c

A parabola is the set of all points in a plane that are the same distance from the focus of the parabola and the directrix.

The focal length is the distance between the vertex and the focus of the parabola. c = focal length

c c

Vertical Parabola Vertex (0, 0) Equation y = ax²

The coefficient a = and c =

focus (0, c) directrix y = – c.

c c

Horizontal Parabola Vertex (0, 0) Equation x = ay²

The coefficient a = and c =

focus (c, 0) directrix x = – c.

Example 1: Determine both the focus and the directrix of each parabola.

a.

b.

a = 2, c = 1/8 , focus (0, 1/8) and directrix y = – 1/8

a = –1/6, c = –3/2 , focus (–3/2, 0) and directrix x = 3/2

Example 2: What is an equation of the parabola with vertex (0, 0) and focus (0, – 1.5)?

Sketch the vertex and the focus to see that the focus is below the vertex, so the parabola will open down. This is a vertical parabola.The focus (o, c), so c = – 1.5 and a = 1/– 6.The equation y = ax² , so y = (–1/6)x².

Example 3: What is an equation of the parabola with vertex at the origin and directrix x = ?

Sketch the vertex and the directrix to see that the directrix is left of the vertex, so the parabola will open to the right. This is a horizontal parabola.The directrix is x = – 5/2 so c = 5/2 and a = 1/10.The equation x= ay² , so x = (1/10)y².

= a and = c

Example 4: What are the vertex, focus, and directrix of the parabola with equation y = x² + 8x + 18?

b = 8c = (8/2)² = 16

y = (x² + 8x + 16) + 18 – 16y = (x + 4)(x + 4) + 2y = (x + 4)² + 2

h = – 4 and k = 2, Vertex: (h, k), so (– 4, 2) a = 1, so c = 1/4Focus: (h, k + c), so (– 4, 2.25)Directrix: y = k – c, so y = 1.75

Example 5: What is the equation of the parabola with vertex (10, 2) and focus (10, 1)?

Sketch the vertex and the focus to see that the focus is below the vertex, so the parabola will open down. This is a vertical parabola.The vertex (h, k), so h =10 and k = 2.The focus (h, k + c), so c = –1 and a = – ¼ .The equation y = a(x – h)² + k , so y = – ¼(x – 10)² + 2 .

Example 6: The mirrored reflector of a flashlight is 16 cm across and 10 cm deep. How far from the vertex should the light bulb be positioned?

Sketch the parabola with the vertex at the origin and have it open up. Use the equation y = ax², then select a point (8, 10) on the graph and substitute those values into the equation x = 8 and y = 10, (10) = a (8)² Solve for a, so a = 5/32 and c = 8/5 = 1.6. The light bulb should be at the focus of the parabola, so it should be 1.6 cm from the vertex point.

Parabola Guided Practice

Parabola Guided PracticeAnswers

AssignmentPH Parabola Worksheet, 1-30 all