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
4
x py

6. Parabolas
 Likewise, the parabola with focus p, 0 and directrix
x = –p has the equation

2
4
y px

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

8. Parabolas
 Example: Find the focus and directrix of the parabola
whose equation is
Focus: 0, 3
Directrix: y = –3

2
12
x y
4 12
p 
3
p

2
4
x 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
4
x h p y k    
  
2
4
y k p x h

10. Parabolas
 Example: Identify the vertex, focus, directrix, and axis of
symmetry for the parabola.
   
  
2
4 8 1
x y

11. Parabolas
 Example: Identify the vertex, focus, directrix, and axis of
symmetry for the parabola.
   
  
2
4 8 1
x y

4 8
p
2
p
vertex: 4, ‒1
(opens vertically) focus:   
1 2 1
4, 1
directrix:
axis of symmetry:
    
1 2 3
y
 4
x

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
4
y 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 1
y x
   
   
2
3 8 1
y x

14. Classwork
 10.1 Assignment (College Algebra)
 Page 957: 8-18 (even); page 906: 68-76 (even);
page 895: 56-60 (even)
 10.1 Classwork Check
 Quiz 9.6