10-5 Parabolas
Objectives
Write the standard equation of a parabola and its axis
of symmetry.
Graph a parabola and identify its focus, directrix, and
axis of symmetry.
Vocabulary
directrix and focus of a parabola
Holt Algebra 2
10-5 Parabolas
A parabola is the set of all
points P(x, y) in a plane that
are an equal distance from
both a fixed point, the
focus, and a fixed line, the
directrix. A parabola has a
axis of symmetry
perpendicular to its directrix
and that passes through its
vertex. The vertex of a
parabola is the midpoint of
the perpendicular segment
connecting the focus and the
directrix.
Holt Algebra 2
10-5 Parabolas
Previously, you have graphed parabolas with
vertical axes of symmetry that open upward or
downward. Parabolas may also have horizontal
axes of symmetry and may open to the left or
right.
The equations of parabolas use the parameter p.
The |p| gives the distance from the vertex to
both the focus and the directrix.
Holt Algebra 2
10-5 Parabolas
Holt Algebra 2
10-5 Parabolas
Notes
1. Write an equation for the parabola with vertex
V(0, 0) and directrix y = 1.
2. Write an equation for the parabola with focus
F(0, 0) and directrix y = -1.
3. Write an equation for the parabola with focus
F(2, -3) and directrix x = 1.
4. Find the vertex, value of p, axis of symmetry, focus,
and directrix of the parabola y – 2 = 1 (x – 4)2,
12
then graph.
Holt Algebra 2
10-5 Parabolas
Example 1A: Writing Equations of Parabolas
Write the equation in standard form for the parabola.
Step 1 Because the axis of
symmetry is vertical and the parabola
opens downward, the equation is in
the form
1
y=
4p
x2 with p = 5.
Step 2: Because p = 5 and the parabola opens y = –
downward. The equation of the parabola is
Holt Algebra 2
1
20
x2
10-5 Parabolas
Example 1B: Writing Equations of Parabolas
Write the equation in standard form for the
parabola with vertex (0, 0) and directrix x = -6
Step 1 Because the directrix is a vertical
line, the equation is in the form
and
the graph opens to the right.
Step 2 The equation of the parabola is x =
Holt Algebra 2
1
y2.
24
10-5 Parabolas
Example 2A: Write the Equation of a Parabola
Write the equation of a parabola with focus F(2, 4) and
directrix y = –4.
Step 1 The vertex is (2,0)… halfway between
the focus and directrix.
Step 2 The equation opens up so
and p = 4.
Step 3
Holt Algebra 2
y=
1
4p
x2
10-5 Parabolas
Example 2B
Write the equation of a parabola with focus
F(0, 4) and directrix y = –4.
Holt Algebra 2
10-5 Parabolas
Example 2C
Write the equation in standard form for each
parabola.
vertex (0, 0), focus (0, –7)
The equation of the parabola is
Holt Algebra 2
10-5 Parabolas
Example 3: Graphing Parabolas
Find the vertex, value of p, axis of
symmetry, focus, and directrix of the
1
parabola y+3 = 8
(x – 2)2. Then graph.
Step 1 The vertex is (2, –3).
Step 2
Holt Algebra 2
1
= 1
4p
8
, so 4p = 8 and p = 2.
10-5 Parabolas
Example 3 Continued
Step 3 The graph has a vertical axis of symmetry,
with equation x = 2, and opens upward.
Step 4 The focus is (2, –3 + 2),
or (2, –1).
Step 5 The directrix is a
horizontal line
y = –3 – 2, or
y = –5.
Holt Algebra 2
10-5 Parabolas
Example 4
Find the vertex, value of p, axis of symmetry, focus, and
directrix of the parabola. Then graph.
Step 1: The vertex is (1, 3)
and p = 3
Step 2 The graph opens right,
with horizontal axis of
symm. y = 3.
Step 3 The directrix is vertical
line x = –2.
Step 4 The focus is (1 + 3, 3),
or (4, 3).
Holt Algebra 2
10-5 Parabolas
Example 5
Find the vertex, value of p axis of symmetry,
focus, and directrix of the parabola. Then graph.
Step 1: The parabola opens
downward, the vertex is (8, 4)
with axis of symm: x = 8.
Step 2: p = ½
Step 3: Use p to find F (8, 3.5)
and directrix: y = 4.5
Holt Algebra 2
10-5 Parabolas
Notes
1. Write an equation for the parabola with vertex
V(0, 0) and directrix y = 1.
2. Write an equation for the parabola with focus
F(0, 0) and directrix y = -1.
3. Write an equation for the parabola with focus
F(2, -3) and directrix x = 1.
4. Find the vertex, value of p, axis of symmetry, focus,
and directrix of the parabola y – 2 = 1 (x – 4)2,
12
then graph.
Holt Algebra 2