Today in Precalculus• Go over homework• Notes: Parabolas with vertex other

than (0,0)• Homework

When a parabola with vertex (0,0) is translated horizontally h units and vertically k units the vertex becomes (h, k)

The translation does NOT change the focal length, the focal width, or the direction the parabola opens.

x

y

x

y

Parabolas with vertex (h,k)

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

Opens Up if p>0

Down if p<0

To right if p>0

To left if p<0

Focus (h, k+p) (h+p, k)

Directrix y = k – p x = h – p

Axis x = h y = k

Focal length p p

Focal width |4p| |4p|

Graphing a parabola by handLet the focus F of a parabola be (2, -3) and its

directrix be y = 4.

Sketch and label the focus and directrix of the parabola.

directrix

F

x

y

x

y

Graphing a parabola by handLocate, sketch, and label the axis of the parabola

What is its equation?

x=2

Label and plot the vertex V of the

parabola. Label it by name and coordinates.

directrix

F

axis

V(2,0.5)

x

y

x

y

x

y

Graphing a parabola by handWhat are the focal length and width of the parabola?

focal length = p = -3.5

focal width =|4(-3.5)|=14

directrix

F

axis

V(2,0.5)

x

y

Use the focal width to locate, plot, and label the endpoints of a chord of the parabola that parallels the directrix.

Sketch the parabola.

Which direction does it open?

downward

What is its equation in standard form?

(x – 2)2 = 4(-3.5)(y – .5)

(x – 2)2 = -14(y – .5)

V(2,0.5)

directrix

F

axis

(-5,-3) (9,-3)

x

y

x

y

Graphing a parabola with the calculator

(x – 2)2 = -14(y – .5)

Solve for y

(x – 2)2 = -14y + 7

(x – 2)2 – 7 = -14y

212 7

14x y

Graphing a parabola with the calculator(y – 3)2 = 6(x – 4)

Solve for y

3 6( 4)y x

6( 4) 3y x

1

2

6( 4) 3

6( 4) 3

y x

y x

Example

Focus (-5,3) and vertex (-5,6)

So p=3, opens downward

(x + 5)2 = 4(3)(y – 3)

(x + 5)2 = 12(y – 3)

Example

Vertex (3,5) and directrix y = 7

So opens downward, p = –2

(x – 3 )2 = 4(-2)(y – 5)

(x + 5)2 = -8(y – 3)

Example

Vertex (-3,3), opens right, focal width = 20

(y – 3 )2 = 20(x +3)

Homework

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