Parabolas

Finding the Rule and Graphing

DefinitionsDefinitions

The parabola has a focus and a directrix l,which is the line NOT passing through the focus

There are four cases according to the concavity

Can have the vertex at the origin, or at (h,k)

d(V,F) = d(V,l) = c

The axis of symmetry passes through the vertex and the focus. It is perpendicular to the directrix.

4 Cases - Centered at Origin4 Cases - Centered at Origin

xx22 = 4cy = 4cy F(0,c)

l: y= - c

4 Cases - Centered at Origin4 Cases - Centered at Origin

xx22 = -4cy = -4cy

F(0,-c)

l: y= c

4 Cases - Centered at Origin4 Cases - Centered at Origin

yy22 = 4cx = 4cxF(c,0)

l: x = - c

4 Cases - Centered at Origin4 Cases - Centered at Origin

yy22 = -4cx = -4cxF(-c,0)

l: x = c

ObservationsObservations

When x2, it opens up or down

When y2, it opens left or right

The axis of symmetry is the y-axis

The axis of symmetry is the x-axis

Example 1 – Example 1 – Represent the following:Represent the following:

yy22 = 2x = 2x

F(0.5,0)

l: x = -0.5

Which way are we opening?

To the right (y2 and positive)

What is c?

c = 2/4 = 0.5Check: 4c = 4(0.5) = 2!

What is l?

l: x = -c so x=-0.5

4 Cases - Centered at (h,k)4 Cases - Centered at (h,k)

(x-h)(x-h)22 = 4c(y-k) = 4c(y-k) F(h,c+k)

l: y= k - c

V(h,k)

4 Cases - Centered at (h,k)4 Cases - Centered at (h,k)

(x-h)(x-h)22 = -4c(y-k) = -4c(y-k)

F(h,k-c)

l: y= k+ cV(h,k)

4 Cases - Centered at (h,k)4 Cases - Centered at (h,k)

(y-k)(y-k)22 = 4c(x-h) = 4c(x-h)F(h+c, k)

l: x = h - c

V(h,k)

4 Cases - Centered at (h,k)4 Cases - Centered at (h,k)

(y-k)(y-k)22 = -4c(x-h) = -4c(x-h)F(h-c,k)

l: x = h+c

V(h,k)

Example 2 – Represent:Example 2 – Represent:(x-2)(x-2)22 = 8(y-4) = 8(y-4)

F(2,6)

l: y= 2

V(2,4)

Which way are we opening?UP! (x2 and positive)What are c and F?c = 8/4 = 2F(h, c+k)=(2,6)

What is l?l: y =k-c so y=2

What is the vertex?(2,4)

HOMEWORK – The only way to make these easier is to practice…

WorkbookWorkbookp. 346 #1,2,3,4,5

(intersections like you always do them!)

p. 348-349 #6,7,8,9(for #9, use the table of values to make your parabola more accurate!)

NEXT CLASS: Hand in p.343 #20 on a loose leaf!NEXT CLASS: Hand in p.343 #20 on a loose leaf!We will move on to general form and inequalitiesWe will move on to general form and inequalities