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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) - PowerPoint PPT Presentation
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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
Page 641: 3,4,22-43 odd
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