10-1 Exploring Conic Sections
Hubarth
Algebra II
A conic section is a curve formed by the intersection of a plane and a double cone.
Ex. 1 Graphing a Circle
Graph the equation x2 + y2 = 16. Describe the graph and its lines of symmetry.
Then find the domain and range.
Plot the points and connect
Make a table of values.
them with a smooth curve.
x
y
–4
0
–3 ± 7
± 2.6
–2 ± 2 3
± 3.5
–1 ± 15
± 3.9
0
±4
1 ± 15
± 3.9
2 ±2 3
± 3.5
3 ± 7
± 2.6
4
0
The graph is a circle of radius 4. Its center is at the origin. Every line through the
center is a line of symmetry.
Recall from Chapter 2 that you can use set notation to describe a domain
or a range. In this Example, the domain is {x|–4 ≤ x ≤4}. The range is {y|–4 ≤ y ≤4}.
Ex. 2 Graphing an Ellipse
Graph the equation 9x2 + 4y2 = 36. Describe the graph and the lines of
symmetry. Then find the domain and range.
Make a table of values.
x
–2
–1
0
1
2
y
0
± 2.6
±3
± 2.6
0
Plot the points and connect
them with smooth curves.
The graph is an ellipse. The center is at the origin. It has two lines of
symmetry, the x-axis and the y-axis.
The domain is {x| –2 <
– x <
– 2}.
The range is {y| –3 <
– y <
– 3}.
Ex. 3 Graphing a Hyperbola
Graph the equation x2 – y2 = 4. Describe the graph and its lines of
symmetry. Then find the domain and range.
Make a table of values.
x
y
–5 ± 4.6
–4 ± 3.5
–3 ± 2.2
–2
0
–1
—
0
—
1
—
2
0
3 ± 2.2
4 ± 3.5
5 ± 4.6
Plot the points and connect
them with smooth curves.
The graph is a hyperbola that consists of two branches. Its center is at
the origin. It has two lines of symmetry, the x-axis and the y-axis.
The domain is {x| x <
– –2 or x >
– 2}. The range is all real numbers.
Ex. 4 Identifying Graphs of Conic Sections
Identify the center and intercepts of the conic section. Then find the
domain and range.
The center of the ellipse is (0, 0).
The x-intercepts are (–5, 0) and (5, 0), and the y-intercepts are (0, –4) and
(0, 4).
The domain is {x| –5 <
– x <
– 5}. The range is {y| –4 <
– y <
– 4}.
Ex. 5 Identifying Graphs of Conic Sections
Identify the center and intercepts of the conic section. Then
find the domain and range.
The center of the hyperbola is (0, 0)
The y-intercepts are at (0, 4) and (0, -4),
and there are no x-intercepts.
The domain is real numbers and the
range is 𝑦| 𝑦 ≥ 4 𝑜𝑟 𝑦 ≤ −4
Ex. 6 Match Each Equation with It’s Conic
Determine whether each equation models a circle, and ellipse, or a
hyperbola.
a. 25x2 + 4y2 = 100
b. x2 + y2 = 16
c. 2x2 – y2 = 16
a.
The equation 25x2 + 4y2 = 100 represents a conic section with
two sets of intercepts, (±2, 0) and (0, ±5). Since the intercepts are
not equidistant from the center, the equation models an ellipse.
b.
The equation x2 + y2 = 16 represents a conic section with two
sets of intercepts, (±4, 0) and (0, ±4). Since each intercept is 4
units from the center, the equation models a circle.
c.
The equation 2x2 – y2 = 16 represents a conic section with one
set of intercepts, (±2 2, 0), so the equation must be a hyperbola.
Practice
1. Graph each equation.
a. 9𝑥 2 + 4𝑦 2 = 36
b. 𝑥 2 − 𝑦 2 = 4
c. 𝑥 2 + 𝑦 2 = 16
2. Determine whether each equation models a circle, and ellipse, or a hyperbola.
a. 4𝑦 2 − 36𝑥 2 = 1
b. 𝑥 2 + 𝑦 2 = 100
c. 4𝑥 2 + 36𝑦 2 = 144
Hyperbola
Circle
Ellipse