10.2 Parabolas

By: L. Keali’i Alicea

Parabolas

• We have seen parabolas before. Can anyone tell me where?

• That’s right! Quadratics!

• Quadratics can take the form:

x2 = 4py or y2 = 4px

Parts of a parabola

• FocusA point that lies on

the axis of symmetry that is equidistant from all the points on the parabola.

Parts of a parabola

• DirectrixA line perpendicular

to the axis of symmetry used in the definition of a parabola.

FocusLies on AOS

Directrix

2 Different Kinds of Parabolas

• x2=4py• y2=4px

Standard equation of Parabola (vertex @ origin)

Equation Focus Directrix AOS

x2=4py (0,p) y=-pVertical

(x=0)

y2=4px (p,0) x=-pHorizontal

(y=0)

x2=4py, p>0

Focus (0,p)

Directrixy=-p

x2=4py, p<0

Focus (0,p)

Directrixy=-p

y2=4px, p>0

Directrixx=-p

Focus (p,0)

y2=4px, p<0

Focus (p,0)

Directrixx=-p

Identify the focus and directrix of the parabola

x = -1/6y2

• Since y is squared, AOS is horizontal• Isolate the y2 → y2 = -6x• Since 4p = -6• p = -6/4 = -3/2

• Focus : (-3/2,0) Directrix : x=-p=3/2• To draw: make a table of values & plot • p<0 so opens left so only choose neg values for x

Your Turn!

• Find the focus and directrix, then graph

x = 3/4y2

• y2 so AOS is Horizontal

• Isolate y2 → y2 = 4/3 x

• 4p = 4/3 p = 1/3

• Focus (1/3,0) Directrix x=-p=-1/3

Writing the equation of a parabola.

• The graph shows V=(0,0)

• Directrex y=-p=-2

• So substitute 2 for p

• x2 = 4py

• x2 = 4(2)y

• x2 = 8y

• y = 1/8 x2 and check in your calculator

Your turn!

• Focus = (0,-3)

• X2 = 4py

• X2 = 4(-3)y

• X2 = -12y

• y=-1/12x2 to check

Assignment10.2 A (1-3, 5-19odd)10.2 A (1-3, 5-19odd)

10.2 B (2-20 even, 21-22)10.2 B (2-20 even, 21-22)