1. 10-1 Exploring Conic Sections
Hubarth
Algebra II

2. A conic section is a curve formed by the intersection of a plane and a double cone.

3. 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 them with a smooth curve.
Make a table of values.
x y –
–3 ± ± 2.6
–2 ± ± 3.5
–1 ± ± 3.9 ±4
1 ± ± 3.9
2 ± ± 3.5
3 ± ± 2.6
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}.

4. Ex. 2 Graphing an Ellipse
Graph the equation 9x y2 = 36. Describe the graph and the lines of symmetry. Then find the domain and range.
x y –
–1 ± 2.6
0 ± 3
1 ± 2.6
Make a table of values.
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| – x }.
The range is {y| – y }.
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5. 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.
–5 ± 4.6
–4 ± 3.5
–3 ± 2.2 –
– — — —
3 ± 2.2
4 ± 3.5
5 ± 4.6
x y
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 }. The range is all real numbers.
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6. 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| – x }. The range is {y| – y }.
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7. 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

8. Ex. 6 Match Each Equation with It’s Conic
Determine whether each equation models a circle, and ellipse, or a hyperbola.
a. 25x y2 = 100
b. x2 + y2 = 16
c. 2x2 – y2 = 16
a. The equation 25x y2 = 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.

9. Practice
1. Graph each equation.
a. 9 𝑥 2 +4 𝑦 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. 4 𝑦 2 −36 𝑥 2 =1 b. 𝑥 2 + 𝑦 2 =100 c. 4 𝑥 𝑦 2 =144
Hyperbola
Circle
Ellipse