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Lesson 10.2 Parabolas Goal: Graph and write equations of parabolas

Lesson 10.2 Parabolas Goal: Graph and write equations of parabolas

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Page 1: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

Lesson 10.2Parabolas

Goal: Graph and write equations of parabolas

Page 2: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

Quadratic Formula

a

acbbx

2

42

02 cbxax

IF

THEN

Page 3: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 4: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 5: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 6: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 7: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 8: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 9: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 10: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

Creation of a Parabola

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

(Zoebel, 1997-2006)

Page 11: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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)

Page 12: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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) 

 

Page 13: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

One of the simplest polynomial functions is

Page 14: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

drops down by three units.

Page 15: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

moves to the right 2 units.

Page 16: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 17: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 18: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 19: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 20: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 21: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

Definition of a Parabola

(Larson, Boswell, Kanold & Stiff, 2005)

Page 22: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

Parabolas

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

pyx 42

Page 23: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

4py

If positive, the parabola opens up

If negative, the parabola opens down

Page 24: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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.

Page 25: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

The Focus

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

Page 26: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

The Directrix

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

Page 27: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 28: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

The Vertex

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

Page 29: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

Definition of a Parabola

(Larson, Boswell, Kanold & Stiff, 2005)

Page 30: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

#32 Identify the focus and directrix of the parabola. 

x y2 8x y2 8

Page 31: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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(

Page 32: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas

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

Page 33: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 34: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas
Page 35: Lesson 10.2 Parabolas  Goal: Graph and write equations of parabolas