Transcript Document

Concept Category 15
Polar Equations & Graphs
LT 6B:
I can represent complex numbers on the
complex plane in rectangular form and polar
form. I can explain why the rectangular and
polar forms of a given complex number
represent the same number. I can graph polar
equations using technology
REASONING Question that comes to
mind
Find the equation of this graph
I. Polar Coordinate System
A. Definition
A system of coordinates in which the location of a point is
determined by its distance from a fixed point at the
center of the coordinate space (called the pole) and by
the measurement of the angle formed by a fixed line (the
polar axis, corresponding to the x -axis in Cartesian
coordinates) and a line from the pole through the given
point. The polar coordinates of a point are given as ( r, θ),
where r is the distance of the point from the pole, and θ
is the measure of the angle.
B. Visual
Polar Angle:
The angle formed by the polar axis
and the radius vector in a polar
coordinate system.
Polar Graph Coordinate Graph
C. Process
Coordinate: (r,Θ)
Equation: r=cosΘ
Types of Polar Graphs:
Circles and Spirals
Types of Polar Graphs:
Limacons
Types of Polar Graphs:
Rose Curves
Types of Polar Graphs:
Leniscates
An interesting spiral….
Relationship Between Polar &
Rectangular Coordinates
• Polar to Rectangular Coordinates
x = rcosθ and y = rsinθ
• Rectangular to Polar
r2 = x2 + y2 and tan = y/x (x≠0)
C. Process
Sketch the graph of
r=cosΘ in the
Polar
Coordinate
System
E. Application
How could we use a logarithmic spiral?
https://youtu.be/3tNVOhtvPEw
Automatic Lawnmower!
You have a perfectly circular lawn in your yard. You would like to devise a method to have
your lawn mower automatically mow that section of your yard. You need to give your
software a polar equation to map out the position of the lawnmower over time. The lawn is
125 square meters and your lawn mower has a 320 mm cut width. Create an equation to
model the best path of the lawn mower as it mows your grass. Use technology to graph and
check your model. How many rotations will the lawn mower need to make in your model?
Goal Problems
SAT 2, Level 2 Prep Book:
Page 69
Practice
Recall & Reproduction:
Pg. 597 #1, #3, #13
Routine:
Pg. 598 #23, 25, 29, 35, 39, 41
Non-Routine:
Pg. 617 #23 (see example 3, page 612)
Pg. 617 #17 (see example 4, page 616)
Parametric Equations
LT 6B
I can analyze whether the parametric
(parameter is time) or rectangular
system is the appropriate choice to
model a given situation.
Fundamental
I can graph and make sense of
Skill parametric
equations. I can model real
world situations with parametric
equations.
Diagram what is happening
What is the difference?
Why is one more challenging
than the other?
Problem: Find Mathematical Model
How can we graph this scenario?
Why care?
I. Parametric
A. Definition: Parametric equations are a set of
equations that express a set of quantities as
explicit functions of a number of
independent variables, known as
"parameters.”
B. Visual
X(t)= 4t2
Y(t)= 3t
Set up a table:
t
X(t)
Y(t)
C. Process
What do you notice?
Polar Connection?
Example:
Given: x(t)=4t and y(t)= t2
How do you eliminate the parameter?
Eliminating Parameters from Parametric Equations:
Given the following parametric equations, can you create
one equivalent equation without the parameter?
Goal Problems
1. The graph defined by the parametric equations
x = cos2t
y = 3 sint -1
is :
A) a circle
B) a hyperbola
C) a vertical line
D) part of a parabola
E) an ellipse
2. A line has parametric equations
x=5+t
y=7+t
where t is the parameter. The slope of the line is:
A)
D)
B) 1
C)
E) 7
Solutions to Goal Problems
1. D
2. B
D. Purpose: applications that involve time as a function of two other variables
1. Nolan Ryan throws a baseball with an initial speed of 145 ft. per
second at an angle of 20° to the horizontal. The ball leaves Nolan
Ryan’s hand at a height of 5 ft.
a. Create an equation or a set of equations that describe
the position of the ball as a function of time.
b. How long is the ball in the air? (Assume the ball hits the
ground without being caught)
c. How far horizontally would the ball travel in the
situation described in (b)?
d. When is the ball at its maximum height? Determine the
maximum height of the ball.
e. Graph the equations to check your answers. Sketch the
graph and show the window
f. Consider the situation present in a game. Nolan Ryan
would like the ball to land in the catcher’s mitt (18 ft.
from him on the mound) within the strike zone. Assume
the strike zone is between 1.5 ft above the ground (knee
height) and 3.75 ft above ground (chest height). If he
maintains his 145 ft. per second velocity on each pitch,
what angle would ensure his pitch hits the very bottom
of the strike zone?
Solution
a)
b) 3.197 seconds
c) 435.6 ft.
d) 43.44 ft at 1.55 seconds
f) Angle: -9.86° or 350.12°
Active Practice
Concept Category #16 (6.3): Parametric Equations & Graphs (Honors)
Recall & Reproduction
Routine
Non-Routine
§10.2 p590 #17-24
§10.2 p590 #25-28
§10.2 p590 #29-30