Warmup Alg 2 19 Apr 2012

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• Section 9.2: Parabolas again!• Non-Zero Vertex

• Completing the Square with Parabolas

Go over assignment from last class period

Section 9.2: Graphing a Parabola with a non-zero

vertex

Vocabulary

Parabola

Focus

Directorix

Vertex

Axis of symmetry

A function with a SINGLE “squared” term

Focus

Directorix

Vertex

Axis of Symmetry

Distances are the same!

Non-Zero Standard equation

Standard Form Vertex Focus Directrix

Vertical (x - h)2 = 4p(y - k) (h, k) (h, k + p) y = k - p

Horizontal (y - k)2 = 4p(x - h) (h, k) (h + p, k) x = h - p

Every point on a parabola is the same distance from the focus as from the directrix

What it looks like

(x - h)2 = 4p(y - k)

What it looks like

(y - k)2 = 4p(x - h)

Graphing

(y - 3)2 = 16(x + 2) Divide by 12 & find “p”

(y - 3)2 = (x + 2) So, p = 3

Vertex is (-2, 3)

Focus is (-2+4, 3)

Why?

Why?

Directrix is x = -2 – 4or x = -6

Why?

Vertex is (-2, 3)

Focus is (2, 3)

Directrix is x = -6

Graphing

Divide by 20 & find “p”

(x + 4)2 = (y + 2) So, p = 5

Vertex is (-4, -2)

Focus is (-4, -2+5)

Why?

Why?

Directrix is y = -2 – 5or y = -7

Why?

(x + 4)2 = 20(y + 2)

Graphing

Vertex is (-4, -2)

Focus is (-4, 3)

Directrix is y = -7

Simplest form

All the equation does is translate the graph.

Left or right is the number next to the “x”

Up or down is the number next to the “y”

But the sign changes! Keep it simple.

Completing the square

y2 – 10y + 5x + 57 = 0

We need to turn this into the standard form!

Recall from back in Chapter 4, the method we used called Completing the Square.

Patterns in the “Genius Way”

x2 + 6x + 9

x2 + 8x + 16  

x2 + 10x + 25  

 

(x+3)2 

(x+4)2 

(x+5)2 

(x-7)2 x2 - 14x + 49

x2 - 20x + ___ (x-__)2 10 100 

x2 - 16x + ___ (x-__)2  8 64 

x2 + bx + ___ (x+__ )2 b/2 (b/2)2 

x2 + 7x + ___ (x+__)2 7/2 49/4 

Completing the square

y2 – 10y - 5x + 55 = 0

We take the “-10” (because the y is squared), divide by 2, and square the answer.

-10/2 = -5

(-5)2 = 25

Completing the square

y2 -10y -5x +55 = 0Our genius numbers are -5 and 25

+5x – 55 +5x - 55 Move stuff

y2 -10y = 5x - 55 Use the 25 to both +25 +25

y2 -10y +25 = 5x - 30 Now we can factor

(y - 5)2 = 5(x – 6)

Vertex is (6, 5)

Focus is (6+5/4, 5)Directrix is x = 6 - 5/4

p = 5/4 (why?)

You Try!

y2 +8y -3x + 22 = 0Our genius numbers are 4 and 16

+3x – 22 +3x - 22 Move stuff

y2 +8y = 3x -22 Use the 16 to both+16 +16

y2 +8y +16 = 3x - 6 Now we can factor

(y +4)2 = 3(x – 2)

Vertex is (-4, 2)

Focus is (-4+3/4, 2)Directrix is x = 6 - 3/4

p = 3/4 (why?)

You Try – Last one!

x2 +12x +8y -20 = 0Our genius numbers are 6 and 36

-8y +20 -8y +20 Move stuff

x2 +12x = -8y +20 Use the 36 to both+36 +36

x2 +12x +36 = -8y + 56 Now we can factor

(x +6)2 = -8(y – 7)

Vertex is (-6, +7)

Focus is (-6, +7-2)Directrix is x = 7 - -2

p = -2 (why?)

Assignment

Section 9.2: Handout