Identify the vertex, directrix, focus, and axis of a parabola Write the equation of a parabola in vertex form

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)