Warm upFind the coordinates of the

center, the foci, vertices and the equation of the asymptotes for

136

)3(

64

)2( 22

yx

Lesson 10-5 ParabolasTo use and determine the standard and general forms of the equation of a parabolaTo graph parabolas

A parabola is the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. As a result, a parabola is the set of points P for which d(F, P) = d(P, D)

Let's sort out this definition by looking at a graph:

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

focus

directrix

Take a line segment perpendicular to the directrix and intersect with a line segment from the focus of the same length. This will be a point on the parabola and will be the same distance from each.

by symmetry we can get the other half

xxy 12342

Based on this definition and using the distance formula we can get a formula for the equation of a parabola with a vertex at the origin that opens left or right

axy 42

a is the distance from the vertex to the focus (or opposite way for directrix)

If the coefficient on x is positive the parabola opens to

the right

If the coefficient on x is negative the parabola opens to the left

2 -7 -6

-5 -4 -3

-

2 -1 1 5 7 3 0 4 6 8

aa

The equation for the parabola shown is:

The parabola opens to the right and the vertex is 3 away from the focus.

For a parabola with the axis of symmetry parallel to the y-axis and vertex at (h, k), the standard form is …

• The equation of the axis of symmetry is x = h.

• The coordinates of the focus are (h, k + p).• The equation of the directrix is y = k - p.

• When p is positive, the parabola opens upward.

• When p is negative, the parabola opens downward.

(x - h)2 = 4p(y - k)

The Standard Form of the Equation with Vertex (h, k)

For a parabola with an axis of symmetry parallel to the x-axis and a vertex at (h, k), the standard form is:

• The equation of the axis of symmetry is y = k.

• The coordinates of the focus are (h + p, k).

• The equation of the directrix is x = h - p.

(y - k)2 = 4p(x - h)

• When p is negative, the parabola opens to the left.

• When p is positive, the parabola opens to the right.

The Standard Form of the Equation with Vertex (h, k)

Our parabola may have horizontal and/or vertical transformations. This would translate the vertex from the origin to some other place. The equations for these parabolas are the same but h is the horizontal shift and k the vertical shift:

kyahx 42

kyahx 42

hxaky 42 hxaky 42

opens up

The vertex will be at (h, k)

opens down

opens right opens left

(-1, 2)

Let's try one: 182 2 xyOpens? y is squared and 8

is positive so right.

Vertex?It is shifted to the left

one and up 2 (set (x + 1) = 0 and get

x = -1 and set (y - 2) = 0 and get

y = 2). Vertex is (-1, 2)

Focus? 4a = 8 so a = 2. Focus is 2 away from vertex in direction parabola opens.

(1, 2)

This line segment parallel to the directrix thru the focus is number in front of parenthesis which is 8, so 4 each way from focus.

(1, 6)

(1, -2)Directrix?

"a" away from the vertex so 2 away in opposite direction of focus.

x = -3

The equation we are given may not be in standard form and we'll have to do some algebraic manipulation to get it that way.

022 xyySince y is squared, we'll complete the square on the y's and get the x term to other side.

______22 xyy

middle coefficient divided by 2 and squared

1 1

must add to this side too to keep equation =

11 2 xy

factor

Now we have it in standard form we can find the vertex, focus, directrix and graph.

(-3/4, -1)

11 2 xy Opens? Right (y squared & no negative)

Vertex?

opposites of these values

(-1, -1)

Focus? 4a = 1 so a = 1/4

segment thru the focus

1

1, so 1/2 each way

Since the focus was at (-3/4, 1), to get the ends of the segment, we'd need to increase the y value of the focus by 1/2 and then decrease the y value by 1/2. (look at the picture to determine this).

(-3/4, -1/2)

(-3/4, -3/2) Directrix? 1/4 away from vertex

x = -5/4

PracticeFor the equation , find

the coordinates of the focus and the vertex and the equations of the directrix and the axis of symmetry.

022 xy

PracticeGraph the equation of the

parabola 022 xy

PracticeWrite the equation in standard

form. 01882 yxx

Light or sound waves collected by a parabola will be reflected by the curve through the focus of the parabola, as shown in the figure. Waves emitted from the focus will be reflected out parallel to the axis of symmetry of a parabola. This property is used in communications technology.

There are many applications that involve parabolas. One is paraboloids of revolution. This is taking a parabola and revolving it to form "a dish".