Algebra 2 C Chapter 10 Conic Sections

Algebra 2 C
Chapter 10
Conic Sections

Section 2
Parabolas

Conic Sections are
created when a plane
intersects a double
cone

We will talk about one per
Section

10.2 Parabolas
10.3 Circles
10.4 ellipses

10.5 Hyperbolas
FYI these are not 2 parabolas

parabolas are slices that don’t
cut through both cones

1

By changing the angle and location of intersection, we
can produce a circle, ellipse, parabola or hyperbola;
or in the special case when the plane touches the
vertex: a point, line, or 2 intersecting lines.

Examples of parabolas
Head lights,

satellite dishes, Projectile motion,
suspension bridges

2

A Parabola is a set of points that is the same
distance from a given line (directrix) and a given
focus point

In short: AB BC
(2, 3) focus

A
B

(x, y)

directrix

C

(x, -1)

Notice the vertex is the midpoint between the directrix and the focus.

The standard form of the equation y -axis Axis of
of a Parabola with vertex (h, k) symmetry
and an axis of symmetry, x = h is
x= h
y a(x h)2 k
(h, k)
•If a > 0 , k is a minimum value vertex
and the parabola opens up.
•If a < 0 , k is a maximum value x -axis
and the parabola opens down.
directrix

y 0.25( x 2) 7Given a vertex ( -2, 7) and a = 0.252

write the equation in standard form

3

board examples

1. Write each equation in standard form. Identify the
vertex, axis of symmetry, and direction it opens

y x2 14x 20

Complete the square y x 7 2 29

(-7, -29) vertex
x = -7 axis of symmetry

Opens up

#2 y 2 x2 6x 12 Opens up
3 x = 4.5 axis of symmetry
2x 12 3 (4.5, -1.5) vertex
y 3 4

22

The latus rectum is a line Axis of
segment perpendicular to the line symmetry
of symmetry and passes through
the focus of the parabola.

The end points are Focus h, k 1
on the parabola. 4a

The length of the 1 Latus Rectum
latus rectum is a V(h, k)

y ax h2 k directrix

yk 1
4a

The shape and distance between the focus and directrix depend on a in the equation
Go back to previous and find the a

4

Write the equation in standard form.

y x2 2x 3

x 1Identify the axis of symmetry

directrix y the focus 1, 3.75

4.25 1,4

vertex updowonr down ?

opens

y 1x 12 4

Section10.2
open left and right parabolas

Ok all of this is still the case for parabolas
That open left verse right

but the equation is an X =

x a(y k)2 h

5

x a(y k)2 h Axis of symmetry
y=k
directrix x h 1
4a Focus h 1 , k
4a
Do you see the changes??

Latus Rectum

a > 0 opens right V(h, k)
a < 0 open left

Relax all of this is on the next slide!!

6

Identify the vertex, focus, axis of symmetry, directrix,
direction it opens, and length of the latus rectum.
And then graph the parabola

x 3( y 1)2 3

Vertex: ( -3, -1)

focus: 211, 1
12

directrix : x 31
12
Axis : y 1

Opens to the right

Latus rectum is 1/3 units

y .00046x2 325

7

More board examples
1. Write the equation for the parabola given the following

information.

Vertex (6, 8) and focus (6, 2)

y ax 62 8

Draw a sketch

Since the vertex is the midpoint
of the directrix and focus we
know the parabola opens down.
So the a is negative.

Further more the directrix is horizontal and is y = 14
why??

Notice a will equal -1/24

1. Write the equation for the parabola given the following

information. Vertex (6, 8) and focus (6, 2)

y ax 62 8

We still need to find the a,
but how??

Back to what we did in the first
example yesterday with 2 equal distances…

and the distance formula 2

y 1 x 6 8OR ….. Go to the next slide
24will find this in a moment.

8