Topics in Analytic Geometry Part II

Transcription

1 Name Chapter 9 Topics in Analytic Geometry Part II Section 9.4 Parametric Equations Objective: In this lesson you learned how to evaluate sets of parametric equations for given values of the parameter and graph curves that are represented by sets of parametric equations and how to rewrite sets of parametric equations as single rectangular equations and find sets of parametric equations for graphs. Important Vocabulary Parameter I. Plane Curves If f and g are continuous functions of t on an interval I, the set of ordered pairs (f(t), g(t)) is a(n) How to evaluate sets of parametric equations for given values of the parameter C. The equations given by x = f(t) and y = g(t) are for C, and t is the. II. Sketching a Plane Curve One way to sketch a curve represented by a pair of parametric equations is to plot points in the. Each set of coordinates How to graph curves that are represented by sets of parametric equations (x, y) is determined from a value chosen for the. By plotting the resulting points in the order of increasing values of t, you trace the curve in a specific direction, called the of the curve. III. Eliminating the Parameter Eliminating the parameter is the process of: How to rewrite sets of parametric equations as single rectangular equations by eliminating the parameter Describe the process used to eliminate the parameter from a set of parametric equations. 1

2 When converting equations from parametric to rectangular form, it may be necessary to alter: To eliminate the parameter in equations involving trigonometric functions, try using the identities: IV. Finding Parametric Equations for a Graph Describe how to find a set of parametric equations for a given graph. How to find sets of parametric equations for graphs 2

3 Section 9.4 Examples Parametric Equations ( 1 ) Sketch the curve described by the parametric equations x = t 3 and y = t 2 + 1, 1 t 3. ( 2 ) Identify the curve represented by the equations x = 1 t+1 and y = t t+1. ( 3 ) Find a set of parametric equations to represent the graph of y = 1 x 2 using the parameters (a) t = x and (b) t = 1 x. 3

4 Section 9.5 Polar Coordinates Objective: In this lesson you learned how to plot points in the polar coordinate system and convert equations from rectangular to polar form and vice versa. I. Introduction To form the polar coordinate system in the plane, fix a point O, called the or, and construct from O an How to plot points and find multiple representations of points in the polar coordinate system initial ray called the. Then each point P in the plane can be assigned as follows: 1) r = 2) θ = In the polar coordinate system, points do not have a unique representation. For instance, the point (r, θ) can be represented as or, where n is any integer. Moreover, the pole is represented by (0, θ), where θ is. II. Coordinate Conversion The polar coordinates (r, θ) are related to the rectangular coordinates (x y) as follows: How to convert points from rectangular to polar form and vice versa III. Equation Conversion To convert a rectangular equation to polar form, you: How to convert equations from rectangular to polar form and vice versa 4

5 Section 9.5 Examples Polar Coordinates ( 1 ) Plot the point (r, θ) = ( 2, 11π ) on the polar coordinate system. 4 ( 2 ) Find another polar representation of the point (4, π 6 ). ( 3 ) Convert the polar coordinates (3, 3π ) to rectangular coordinates. 2 ( 4 ) Find the rectangular equation corresponding to the polar equation r = 5 sin θ. 5

6 Section 9.6 Graphs of Polar Equations Objective: In this lesson you learned how to graph polar equations. I. Introduction The graph of the polar equation r = f(θ) can be rewritten in parametric form, using t as a parameter, as follows: How to graph polar equations by point plotting II. Symmetry The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation. Substitution 1) The line θ = π 2 : How to use symmetry as a sketching aid 2) The polar axis: 3) The pole: III. Zeros and Maximum r-values Two additional aids to sketching graphs of polar equations are: How to use zeros and maximum r-values as sketching aids 6

7 IV. Special Polar Graphs List the general equations that yield each of the following types of special polar graphs: How to find sets of parametric equations for graphs Limacons: Rose Curves: Circles: Lemniscates: 7

8 Section 9.6 Examples Graphs of Polar Equations ( 1 ) Use point plotting to sketch the graph of the polar equation r = 3 cos θ. ( 2 ) Describe the symmetry of the polar equation r = 2(1 sin θ). ( 3 ) Describe the zeros and maximum r-values of the polar equation r = 5 cos 2θ. 8