* Graph parabolas with vertices at the origin.
* Write equations of parabolas in standard form.
* Graph parabolas with vertices not at the origin.
* Solve applied problems involving parabolas.
2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Graph parabolas with vertices at the origin.
⚫ Write equations of parabolas in standard form.
⚫ Graph parabolas with vertices not at the origin.
⚫ Solve applied problems involving parabolas.
3. 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.
4. 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
5. 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
Vertex: (0, 0)
Focus: (0, 3)
Directrix: y = –3
=
2
12
x y
4 12
p =
= 3
p
8. 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
9. Parabolas
⚫ Example: Identify the vertex, focus, directrix, and axis of
symmetry for the parabola.
( ) ( )
− = +
2
4 8 1
x y
10. 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).
( ) ( )
− = −
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
