This document covers the characteristics, equations, and geometric properties of parabolas in analytic geometry. It explains how to identify the vertex, focus, directrix, and axis of symmetry, as well as how to write the equation of a parabola in vertex form. Additionally, it includes examples and exercises to reinforce understanding of the concepts presented.

1. 10.1 Parabolas
Chapter 10 Analytic Geometry

2. Concepts and Objectives
⚫ Parabolas
⚫ Identify the vertex, directrix, focus and axis of a
parabola
⚫ Write the equation of a parabola in vertex form

3. Parabolas
⚫ The graph of the equation
is a parabola with vertex (h, k) and the vertical line x = h
as axis. It opens up if a > 0 and down if a < 0.
⚫ If we interchange (x – h) and (y – k), we get the equation
which is a parabola with vertex (h, k) and the horizontal
line y = k as axis. It opens to the right if a > 0 and to the
left if a < 0.
( )− = −
2
y k a x h
( )− = −
2
x h a y k

4. Parabolas
⚫ From a geometric standpoint, a parabola is the set of
points in a plane equidistant from a fixed point and a
fixed line. The fixed point is called the focus, and the
fixed line is called the directrix of the parabola.

5. Parabolas
⚫ The parabola has only one squared term, and it opens in
the direction of the nonsquared term.
⚫ The parabola with focus (0, p) and directrix y = –p has
the equation
=2
4x py

6. Parabolas
⚫ Likewise, the parabola with focus (p, 0) and directrix
x = –p has the equation
=2
4y px

7. Parabolas
⚫ Example: Find the focus and directrix of the parabola
whose equation is
=2
12x y

8. Parabolas
⚫ Example: Find the focus and directrix of the parabola
whose equation is
Focus: (0, 3)
Directrix: y = –3
=2
12x y
4 12p =
= 3p
=2
4x py

9. Parabolas
⚫ For a parabola whose vertex is not at the origin, we can
replace the x with (x – h)and y with (y – k):
or
where the focus is distance p from the vertex.
( ) ( )− = −
2
4x h p y k ( ) ( )− = −
2
4y k p x h

10. Parabolas
⚫ Example: Identify the vertex, focus, directrix, and axis of
symmetry for the parabola.
( ) ( )− = +
2
4 8 1x y

11. Parabolas
⚫ Example: Identify the vertex, focus, directrix, and axis of
symmetry for the parabola.
( ) ( )− = +
2
4 8 1x y
=4 8p
=2p
vertex: (4, ‒1)
(opens vertically) focus: − + =1 2 1
(4, 1)
directrix:
axis of symmetry:
= − − = −1 2 3y
= 4x

12. Parabolas
⚫ Example: Write an equation for the parabola with vertex
(1, 3) and focus (–1, 3).

13. Parabolas
⚫ Example: Write an equation for the parabola with vertex
(1, 3) and focus (–1, 3).
( ) ( )− = −
2
4y k p x h
The distance between the focus
and the vertex is p = –1 – 1 = –2,
and the equation is focus vertex
( ) ( )( )− = − −
2
3 4 2 1y x
( ) ( )− = − −
2
3 8 1y x

14. Classwork
⚫ College Algebra & Trigonometry
⚫ Page 957: 8-18 (even); page 199: 20-28 (even);
page 907: 76-80 (even)