Parabolas
Dec 23, 2015
Definitions
• Parabola – set of all points equidistant from a fixed line (directrix) and a fixed point (focus)
• Vertex – midpoint of segment from focus to directrix.
• Axis of Symmetry: line through the focus and vertex
• Vertical Parabola Form: (x – h)2 = 4p(y – k) • Horizontal Parabola form: (y – k)2 = 4p(x – h)
FOCUS FOCUS
VERTEX
VERTEX
DIRECTRIXDIRECTRIX
24x h p y k 2
4y k p x h
VERTICAL PARABOLA HORIZONTAL PARABOLAOpens either up or down. Up if p > 0, down if p < 0.
Opens either left or right. Right if p > 0, left if p < 0.
p: distance from the vertex to the focus and from the vertex to the directrix
p
p
2p 2p
Important note: The points on a parabola are symmetric across the Axis of Symmetry. So, if you know one point, you can find the other.
Focal Chord – the line segment that goes through the focus, and has endpoints on the parabola. It’s length is 4p.
Ex 1: Write the equation of the parabola in standard form.
22 4 8 8 0y x x
24 2 ____ 2 8 ____x x y
24 8 2 8x x y
2 1 44 2 2 8x x y
24 1 2 4x y
24 1 2 2x y
2 11 2
2x y
Decide whether the parabola has vertical or horizontal axis of symmetry, and tell which way the graph opens.
2. -6x2 = 3y
3. (y – 4)2 = 3x + 1
4. 2y = (x + 1)2
Vertical
Opens down
Horizontal
Opens right
Vertical
Opens up
HINT: Which variable is squared?
Given the following information, write the equation of the parabola
5. Vertex (-2, 5); p= -½; Vertical Axis of Symmetry
6. Vertex (1, -3); p = 1/8; Vertical Axis of Symmetry
22 2( 5)x y
2 11 3
2x y 2
4 ( )x h p y k
24 ( )x h p y k
Write the formula, and fill in values
Write the formula, and fill in values
Given the following information, write the equation of the parabola
7. Vertex (6, -1); p = -1/12; Horizontal Axis of Symmetry
8. Vertex (-5, -7); p = 1; Horizontal Axis of Symmetry
Write the formula, and fill in values
Write the formula, and fill in values
27 4 5y x
2 11 6
3y x
24y k p x h
24y k p x h
Vertex:
Focus:
Directrix:
Axis:
Equation:
(0,0)
(0,3)
y = -3
x = 0
What is p?
So, 4p is…?
Is this a horizontal or vertical parabola?
Write the formula, and fill in values.
3
Vertical
4 12p
212y x 212 0 0y x
212y x
24p y k x h
positive or negative?
Why?
9
Vertex:
Focus:
Directrix:
Axis:
Equation:
(0,0)
(-2, 0)
x = 2
y = 0
What is p?
Is this a horizontal or vertical parabola?
Write the formula, and fill in values.
2
Horizontal
4 8p
2 8y x 20 8 0y x
28x y
24y k p x h
positive or negative?
Why?
10
Vertex:
Focus:
Directrix:
Axis:
Equation:
(3,2)
(3,1)
y = 3
x = 3
What is p?
Is this a horizontal or vertical parabola?
Write the formula, and fill in values.
1
Vertical
4 4p
24 2 3y x
24 2 3y x
24p y k x h
positive or negative?
Why?
11
Vertex:
Focus:
Directrix:
Axis:
Equation:
(-4,2)
(0,2)
x = -8
y = 2
What is p?
Is this a horizontal or vertical parabola?
Write the formula, and fill in values.
4
Horizontal
4 16p
216 4 2x y
216 4 2x y
24p x h y k
positive or negative?
Why?
12
13. Find the Vertex, the Focus, the Directrix, and sketch the graph
212 1 3y x
What is the vertex?
What is p?
Which way does the graph open?
Where is the focus?
Where is the directrix?
Sketch the graph.
(3, -1)
3
Up
(3, 2)
y = -4
14. Find the Vertex, the Focus, the Directrix, and sketch the graph
24 3 5x y
What is the vertex?
What is p?
Which way does the graph open?
Where is the focus?
Where is the directrix?
Sketch the graph.
(3, 5)
-1
Left
(2, 5)
x = 4
15. Find the equation if…
The Vertex is (-3, 6), and the Focus is (5, 6)What do we need for the equation?
We need the vertex (GOT IT!) and p.
Draw a sketch.
How far away from the vertex is the focus?
8Positive or Negative? Therefore, p = 8 and 4p = 32So, the equation is:
232 3 6x y
16. Find the equation if…
The Vertex is (2, -1), and the Directrix is x = 5What do we need for the equation?
We need the vertex (GOT IT!) and p.
Draw a sketch.
How far away from the vertex is the directrix?
3Positive or Negative?Therefore, p = -3 and 4p = -12So, the equation is:
212 2 1x y
17. Find the equation if…
The Directrix is y = 5, and the Focus is (-3, 1)What do we need for the equation?
We need the vertex and p.
Draw a sketch. Where is the vertex in relation to the focus and directrix?
Halfway in betweenTherefore, the vertex is at
(-3, 3)Opens up or down?Therefore, p = -2 and 4p = -8So, the equation is:
28 3 3y x
Down
How does a Parabola Work?
Anything entering the parabola is reflected to the focus, concentrating
the signal.
Anything leaving from the focus reflects off the parabola in a straight line creating a
beam.
How does a Parabola Work?