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10.2 Parabolas By: L. Keali’i Alicea

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Parabolas We have seen parabolas before. Can anyone tell me where? That’s right! Quadratics! Quadratics can take the form: x 2 = 4py or y 2 = 4px

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Parts of a parabola Focus A point that lies on the axis of symmetry that is equidistant from all the points on the parabola.

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Parts of a parabola Directrix A line perpendicular to the axis of symmetry used in the definition of a parabola.

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Focus Lies on AOS Directrix

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2 Different Kinds of Parabolas x 2 =4py y 2 =4px

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Standard equation of Parabola (vertex @ origin) EquationFocusDirectrixAOS x 2 =4py(0,p)y=-p Vertical (x=0) y 2 =4px(p,0)x=-p Horizontal (y=0)

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x 2 =4py, p>0 Focus (0,p) Directrix y=-p

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x 2 =4py, p<0 Focus (0,p) Directrix y=-p

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y 2 =4px, p>0 Directrix x=-p Focus (p,0)

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y 2 =4px, p<0 Focus (p,0) Directrix x=-p

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Identify the focus and directrix of the parabola x = -1/6y 2 Since y is squared, AOS is horizontal Isolate the y 2 → y 2 = -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

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Your Turn! Find the focus and directrix, then graph x = 3/4y 2 y 2 so AOS is Horizontal Isolate y 2 → y 2 = 4/3 x 4p = 4/3 p = 1/3 Focus (1/3,0) Directrix x=-p=-1/3

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Writing the equation of a parabola. The graph shows V=(0,0) Directrex y=-p=-2 So substitute 2 for p

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x 2 = 4py x 2 = 4(2)y x 2 = 8y y = 1/8 x 2 and check in your calculator

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Your turn! Focus = (0,-3) X 2 = 4py X 2 = 4(-3)y X 2 = -12y y=-1/12x 2 to check

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Assignment Assignment 10.2 A (1-3, 5-19odd) 10.2 B (2-20 even, 21-22) Assignment

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