lesson 10.2 parabolas goal: graph and write equations of parabolas

Lesson 10.2Parabolas

Goal: Graph and write equations of parabolas

Quadratic Formula

a

acbbx

2

42

02 cbxax

IF

THEN

Objective - To solve quadratic equations using the quadratic formula.

Quadratic Equation Quadratic Formula

y ax2 bx c

x b b2 4ac

2a

Solve.

2x2 5x 3 0

5 52 4 2 3 2 2

5 25 24

4

5 7

4

5 49

4

5 7

4or

5 74

24

or 12

4

12

, 3

a 2 b 5 c 3

Solve 3x2 4x 1 0. Round to the nearest

hundredth.

b b2 4ac2a

-4 -4 2 4 3 -1 2 3

4 16 126

4 286

a 3b 4c 1

6

28+4

6

284

≈ 1.55

≈ −0.22

x ≈ 1.55 or x ≈ -0.22

Solve. 3x2 + 2x – 3 = 0 a = 3b = 2c = -3

b b2 4ac2a

)3(2

)3)(3(4)2(±2 2

6

36+4±2

6

40±2

6

40±2

6

40+26

402

x ≈ 0.72 x ≈ −1.39

Solve. 5x2 -2x – 1 = 0

b b2 4ac2a

a = 5b = -2c = -1

)5(2

)1)(5(4)2()2( 2

10

20+4±2

10

24±2

10

24±2

10

24+210

242

Find the x - intercepts of the graph of

y x2 5x 2. Round to the nearest tenth.

b b2 4ac2a

5 5 2 4 -1 2 2 -1

5 25 8 2

5 33 2

5 33 2

5 33 2

or 5 33

2

a 1b 5c 2

Solve.

x2 3x 7

7 7

x2 3x 7 0

b b2 4ac2a

3 32 4 1 -7 2 1

3 9 282

3 372

3 372

or 3 37

2

a 1b 3c 7

Creation of a Parabola

A conic section is a curve formed by the intersection of a plane a double-napped cone

(Zoebel, 1997-2006)

Where are parabolas?(Internet Access is Required)

• They’re everywhere.

• Put arrow on icon and click.

• Click power point icon on task bar to continue with slide show after video is finished.

(Part 1-They’re Out There!!!, 2008)

When you transform a functionInside the parentheses translates left and right (opposite of what you think)Outside the parentheses translates up and down (exactly what you think) 

One of the simplest polynomial functions is

drops down by three units.

moves to the right 2 units.

This version moves it left 2, up 3, and vertically compresses

Definition of a Parabola

(Larson, Boswell, Kanold & Stiff, 2005)

Parabolas

Parabolas with vertex at (0,0) and open up or down are in the form:

pyx 42

4py

If positive, the parabola opens up

If negative, the parabola opens down

The Axis of Symmetry

For parabolas that open up or down, the axis of symmetry is the line x = the x-coordinate of the vertex.

The Focus

The focus is an ordered pair (x,y), and is INSIDE the parabola and on the axis of symmetry.

The Directrix

The directrix is a line that is perpendicular to the axis of symmetry and is always OUTSIDE the parabola.

4p

4p is the number in front of the variable that has a coefficient of 1.

is the distance from the vertex to the focus and/or the distance from the vertex to the directrix.

p

The Vertex

The vertex lies halfway between the focus ( x, y) and the directrix (line).

Definition of a Parabola

(Larson, Boswell, Kanold & Stiff, 2005)

#32 Identify the focus and directrix of the parabola. 

x y2 8x y2 8

opens up, with vertex at origin, to get the focus, plot the point 2 units inside the parabola and on the axis of symmetry, thus the focus is .

4 8 2p p

( , )0 2

)2,0(

The directrix is perpendicular to the axis of symmetry and is also 2 units away from the vertex, so the equation of the directrix is

y 2

2y