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 them with a smooth curve. Make a table of values. 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. x y –2 0 –1 ± 2.6 0 ± 3 1 ± 2.6 2 0 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| –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. –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 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 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. 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 2 +36 𝑦 2 =144 Hyperbola Circle Ellipse