10.2 Parabolas By: L. Keali’i Alicea. Parabolas We have seen parabolas before. Can anyone tell me...
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Transcript of 10.2 Parabolas By: L. Keali’i Alicea. Parabolas We have seen parabolas before. Can anyone tell me...
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)