Parabolas

Algebra II, Sections 11.1 and 11.5State Standards: Algebra II, Standards 16.0 and 17.0

Mathematical Analysis, Standard 5.1

Definition of a ParabolaA parabola is a collection of all the points that are equidistant from a fixed point known as the focus and a line known as the directrix.

Parts of a Parabola

The parabola

The Focus1/4a units from the vertex

The Directrix1/4a units from the vertex

Latus RectumPasses through the focus

1/a units long

Axis of Symmetry

Recognizing a Parabola: General Form2

2

0

0

Ax Ex Dy F or

Cy Ex Dy F

Either A or C is missing

Standard Forms

2 2

2

2

or Center at (0, 0)

( )

( ) +h Center at (h, k)

y ax x ay

y a x h k or

x a y k

The focus and directix are located 1/(4a) units from the vertex in the direction of the plain (unsquared) variable..

Tip:

y = parabolas open in the y direction. Everything happens to y.

x = parabolas open in the x direction. Everything happens to x

Note the positions of h and k. h goes with x and k goes with y

List of FormulasForm of Equation

Vertical Horizontal 2( )y a x h k

Vertex (h, k)

Focus 1,4

h ka

Directrix 14

y ka

Length of Latus Rectum 1a

Axis of Symmetry x = h

00

aa

Direction

2( )x a y k h

(h, k)1 ,

4h k

a

14

x ha

1a

y = k

00

aa

p = 1/(4a)

Graphing Parabolas

Acceptable• Find the vertex• Find the direction• Make some

acknowledgement of a and the width of the parabola. (i.e. wide or narrow)

Exemplary• Find the vertex• Find the direction• Use a to roughly count the

“slope” of the parabola.• Able to find (when asked)

– Focus– Directrix– Axis of symmetry– Size of latus rectum

• Finds intercepts when practicable.

Graphing ParabolasExample 1

22y x

1. Find the vertex.• If there is nothing in the h or k position,

the coordinate is 0.

• Vertex (0, 0)

2. Find the direction• Since y is plain, it is a vertical parabola.

• Since a > 0, it opens up.

• Since a > 1, it will be somewhat skinny.

• The red points were found by counting “slope.”

3. The focus is 1/(4(2)) units from the vertex (0, 1/8)

4. The directrix is y = -1/(4(2)) = -1/8

5. Axis of symmetry is the y-axis

6. Latus rectum = ½

7. Intercept at (0, 0)

Graphing ParabolasExample 2

2( 3) 2x y

1. Find the vertex.• The variable in the parentheses lets you

know which coordinate you are finding.

• Only numbers in ( ) change signs

• Vertex (2, 3)

2. Find the direction• Since x is plain, it is a horizontal parabola.

• Since a < 0, it opens left.

• Since a = 1, it will be standard width.

• The red points were found by counting “slope.”

3. The focus is 1/(4(1)) units from the vertex (1 ¾, 3)

4. The directrix is x = 2 +1/(4(1)) = 2 ¼

5. Axis of symmetry is the y = 3

6. Latus rectum = 1/4

7. y-intercepts irrational, x-intercept = -7

Graphing ParabolasExample 3

21 ( 2) 14

y x

1. Find the vertex.• The variable in the parentheses lets you

know which coordinate you are finding.

• Only numbers in ( ) change signs

• Vertex (2, 1)

2. Find the direction• Since y is plain, it is a vertical parabola.

• Since a < 0, it opens downward.

• Since a < 1, it will be fat/wide.

• The red points were found by counting “slope.”

3. The focus is 1/(4( ¼)) = 1 unit from the vertex (2, 0)

4. The directrix is y = 1 +1/(4( ¼ )) = 2

5. Axis of symmetry is the x=2

6. Latus rectum = 1/ ¼ =4

7. Intercepts are (0, 0) and (4, 0)

Completing the Square on a Parabola

1. Move the “plain”variable to one side and everything else to the other. Divide by the coefficient of this variable if needed.

2. Group the variables.3. Factor out the “squared” coefficient.4. Complete the square

a) Write down the x (or y) coefficientb) Divide it by 2 (or multiply by ½)c) Square and add the result inside the parentheses. Subtract what you really added

on the endd) Factor as a perfect square trinomial (it’s the middle number)

5. Read your information.

We want to turn this2

2

0

0

Ax Ex Dy F or

Cy Ex Dy F

into

2

2

( )

( ) +h

y a x h k or

x a y k

Example 1 23 18 5 0x x y

1. Move the plain variable (y) to one side and everything else to the other

23 18 5y x x

2. Group the variables 23 18 5y x x

3. Factor out the squared coefficient 23 6 5y x x 4. Complete the square

Write down the coefficient -6

Divide it by 2 -3

Square and add inside the ( ) 9

Subtract what you truly addedon the back 3(9)

Factor as a perfect square trinomial (middle number)

23 6 5 3(9)9y x x

233 22y x

5. Read your info and graph 233 22y x Vertex (3, -22)

Opens upward

Skinny

Example 2 28 2 48 10 0y x y

1. Move the plain variable (x) to one side and everything else to the other. Divide by the x coefficient.

2

2

2 8 48 10

4 24 5

x y y

x y y

2. Group the variables 24 24 5x y y

3. Factor out the squared coefficient 24 6 5x y y 4. Complete the square

Write down the coefficient -6

Divide it by 2 -3

Square and add inside the ( ) 9

Subtract what you truly addedon the back -4(9)

Factor as a perfect square trinomial (middle number)

2 4(4 6 959 )x y y

24 313x y

24 313x y 5. Read your info and graph

Vertex (31, 3)

Opens left

Skinny

Not visible on the graph, but y = 0 gives x =- 5. Intercept (-5, 0)