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§ 10.4
The Parabola; Identifying Conic Sections
Equation of a Parabola
We looked at parabolas in Chapter 8, viewing them as graphs of
quadratic functions. In this section, we return again to the study of
parabolas. Within this chapter on Conics, we extend
our study by considering the formal definition of the parabola.
A parabola is the set of all points in the plane that are equidistant
from a fixed line (called the directrix) and a fixed point (called the
focus).
Note that the definition of both the hyperbola and the ellipse
involved two fixed points, the foci. By contrast, the definition of a
parabola is based on one point and a line.
Blitzer, Algebra for College Students, 6e – Slide #2 Section 10.4
Equation of a Parabola
Definition of a Parabola
A parabola is the set of all points in a plane that are
equidistant from a fixed line (the directrix) and a fixed
point (the focus) that is not on the line.
Axis of
Symmetry
Parabola
Focus
Vertex
Directrix
Blitzer, Algebra for College Students, 6e – Slide #3 Section 10.4
Equation of a Parabola
Parabolas Opening to the Left or to the Right
2
The graphs of x a y k h and x ay 2 by c are parabolas
opening to the left or to the right.
1) If a > 0, the graph opens to the right. If a < 0, the graph opens to
the left.
2
2) The vertex of x a y k h is h, k .
3) The y-coordinate of the vertex of
x ay 2 by c is y
b
.
2a
Blitzer, Algebra for College Students, 6e – Slide #4 Section 10.4
Equation of a Parabola
CONTINUED
y
y
(h, k)
y=k
y=k
(h, k)
x
x a y k h
a0
2
x
x a y k h
a0
Blitzer, Algebra for College Students, 6e – Slide #5 Section 10.4
2
Equation of a Parabola
EXAMPLE
Graph: x y 22 1.
SOLUTION
We can graph this equation by following the steps in the
preceding box. We begin by identifying values for a, k, and h.
x a y k h
2
x y 2 1
2
1) Determine how the parabola opens. Note that a, the
coefficient of y 2 , is 1. Thus, a > 0; this positive value tells us
that the parabola opens to the right.
Blitzer, Algebra for College Students, 6e – Slide #6 Section 10.4
Equation of a Parabola
Graphing Horizontal Parabolas
2
To graph x a y k h or x ay 2 by c,
1) Determine whether the parabola opens to the left or to the right. If a > 0, it
opens to the right. If a < 0, it opens to the left.
2) Determine the vertex of the parabola. The vertex of x a y k 2 h is
b
(h, k). The y-coordinate of the vertex of x ay 2 by c is y .
2a
Substitute this value of y- into the equation to find the x-coordinate.
3) Find the x-intercept by replacing y with 0.
4) Find any y-intercepts by replacing x with 0. Solve the resulting quadratic
equation for y.
5) Plot the intercepts and the vertex. Connect a more accurate graph, select
values for y on each side of the axis of symmetry and compute values for x.
Blitzer, Algebra for College Students, 6e – Slide #7 Section 10.4
Equation of a Parabola
CONTINUED
2) Find the vertex. The vertex of the parabola is at (h, k).
Because k = 2 and h = 1, the parabola has its vertex at (1, 2).
2
3) Find the x-intercept. Replace y with 0 in x y 2 1.
x 0 2 1 2 1 4 1 5
2
2
The x-intercept is 5. The parabola passes through (5, 0).
4) Find the y-intercepts. Replace x with 0 in the given equation.
2
x y 2 1
This is the given equation.
2
0 y 2 1
Replace x with 0.
2
1 y 2
Subtract 1 from both sides.
This equation clearly has no solutions since the left side is a
negative number. Therefore, there are no y-intercepts.
Blitzer, Algebra for College Students, 6e – Slide #8 Section 10.4
Equation of a Parabola
CONTINUED
5) Graph the parabola. With a vertex at (1, 2), an x-intercept at
5, and no y-intercepts, the graph of the parabola is shown as
follows. The axis of symmetry is the horizontal line whose
equation is y = 2.
6
5
Vertex: (1, 2)
Axis of
symmetry:
y = 2.
4
3
2
1
0
-1
-2
0
2
4
6
8
10
x-intercept:
(5, 0)
Blitzer, Algebra for College Students, 6e – Slide #9 Section 10.4
12
Equation of a Parabola
EXAMPLE
Graph: x 2 y 2 4 y 3.
SOLUTION
1) Determine how the parabola opens. Note that a, the
coefficient of y 2, is -2. Thus a < 0; this negative value tells us
that the parabola opens to the left.
2) Find the vertex. We know that the y-coordinate of the vertex
b
is y . We identify a, b, and c in x ay 2 by c.
2a
x 2 y 2 4 y 3
a = -2
b=4
c = -3
Blitzer, Algebra for College Students, 6e – Slide #10 Section 10.4
Equation of a Parabola
CONTINUED
Substitute the values of a and b into the equation for the ycoordinate:
y
b
4
1.
2a
2 2
The y-coordinate of the vertex is 1. We substitute 1 for y into the
parabola’s equation, x 2 y 2 4 y 3 to find the x-coordinate:
x 21 41 3 2 4 3 1.
2
The vertex is at (-1, 1).
3) Find the x-intercept. Replace y with 0 in x 2 y 2 4 y 3.
x 20 40 3 3
2
The x-intercept is -3. The parabola passes through (-3, 0).
Blitzer, Algebra for College Students, 6e – Slide #11 Section 10.4
Equation of a Parabola
CONTINUED
4) Find the y-intercepts. Replace x with 0 in the given equation.
This is the given equation.
x 2 y 2 4 y 3
Replace x with 0.
0 2 y 2 4 y 3
4 42 4 2 3 Use the quadratic formula to solve
y
for y.
2 2
4 16 24
4
4 8
y
4
y
Simplify.
Subtract.
This equation clearly has no solutions since the radicand is a
negative number. Therefore, there are no y-intercepts.
Blitzer, Algebra for College Students, 6e – Slide #12 Section 10.4
Equation of a Parabola
CONTINUED
5) Graph the parabola. With a vertex at (-1, 1), an x-intercept at
-3, and no y-intercepts, the graph of the parabola is shown below.
The axis of symmetry is the horizontal line whose equation is y =
1.
5
Axis of
symmetry:
y = 1.
4
Vertex: (-1, 1)
3
2
1
0
-25
-20
-15
-10
-5
-1
-2
x-intercept:
(-3, 0)
-3
Blitzer, Algebra for College Students, 6e – Slide #13 Section 10.4
0
Equations of Conic Sections
Recognizing Conic Sections from Equations
Conic Section
How to Identify the Equation
Circle
When x - and y 2-terms are on the
same side, they have the same
coefficient.
Ellipse
When x - and y 2-terms are on the
same side, they have different
coefficients of the same sign.
or (dividing by 64)
When x 2- and y 2-terms are on the
same side, they have coefficients
with opposite signs.
or (dividing
by 36)
2
2
Hyperbola
Parabola
Example
2
x 2 y 2 16
4 x 2 16 y 2 64
2
x2 y2
1
16 4
9 y 2 4 x 2 36
Only one of the variables is
squared.
Blitzer, Algebra for College Students, 6e – Slide #14 Section 10.4
y
x
1
4
9
x y2 4 y 5
Equations of Conic Sections
EXAMPLE
Indicate whether the graph of each equation is a circle, an ellipse,
a hyperbola, or a parabola: (a) x 2 36 4 y 2 (b) 3x 2 27 3 y 2 .
SOLUTION
(Throughout the solution, in addition to identifying each
equation’s graph, we’ll also discuss the graph’s important
features.) If both variables are squared, the graph of the equation
is not a parabola. In both cases, we collect the x 2- and y 2-terms
on the same side of the equation.
(a) x 2 36 4 y 2
The graph cannot be a parabola. To see if it is a circle, an ellipse,
2
or a hyperbola, we collect the - and x 2-termsy on
the same side.
Add
to both sides.
4 y2
Blitzer, Algebra for College Students, 6e – Slide #15 Section 10.4
Equations of Conic Sections
CONTINUED
We obtain the following: x 2 4 y 2 36.
Because the x 2- and y 2 -terms have different coefficients of the
same sign, the equation’s graph is an ellipse.
(b) 3x 2 27 3 y 2
The graph cannot be a parabola. To see if it is a circle, an ellipse,
or a hyperbola, we collect the x 2 - and y 2-terms on the same side.
Subtract 3y 2 from both sides. We obtain:
3x 2 3 y 2 27
Because the x 2- and y 2 -terms have coefficients with opposite
signs, the equation’s graph is a hyperbola.
Blitzer, Algebra for College Students, 6e – Slide #16 Section 10.4
The Parabola
A parabola is the set of all points that are equidistant from a
fixed line, the directrix, and a fixed point, the focus, that is
not on the line.
The line passing through the focus and perpendicular to the
directrix is the axis of symmetry. The point of intersection
of the parabola with its axis of symmetry is the vertex.
Blitzer, Algebra for College Students, 6e – Slide #17 Section 10.4