Concept Category 15 Polar Equations & Graphs LT 6B: I can represent complex numbers on the complex...

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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

Transcript of Concept Category 15 Polar Equations & Graphs LT 6B: I can represent complex numbers on the complex...

Page 1: 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.

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

Page 2: 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.

REASONING Question that comes to mind

Page 3: 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.

Find the equation of this graph

Page 4: 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.

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.

Page 5: 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.

B. Visual

Polar Angle:The angle formed by the polar axis and the radius vector in a polar coordinate system.

Page 6: 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.

Polar Graph Coordinate Graph

Page 7: 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.

C. ProcessCoordinate: (r,Θ)Equation: r=cosΘ

Page 8: 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.

Types of Polar Graphs:Circles and Spirals

Page 9: 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.

Types of Polar Graphs:Limacons

Page 10: 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.

Types of Polar Graphs:Rose Curves

Page 11: 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.

Types of Polar Graphs:Leniscates

Page 12: 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.

An interesting spiral….

Page 13: 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.

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)

Page 14: 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.
Page 15: 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.

C. ProcessSketch the graph of r=cosΘ in the Polar Coordinate System

Page 16: 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.

https://youtu.be/3tNVOhtvPEw

How could we use a logarithmic spiral?

E. Application

Page 17: 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.

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?

Page 18: 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.

Goal Problems

SAT 2, Level 2 Prep Book:

Page 69

Page 19: 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.

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)

Page 20: 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.

LT 6B I can analyze whether the parametric (parameter is time) or rectangular system is the appropriate choice to model a given

situation.

Fundamental Skill I can graph and make sense of parametric

equations. I can model real world situations with parametric equations.

Parametric Equations

Page 21: 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.

Diagram what is happeningWhat is the difference?

Why is one more challenging than the other?

Page 22: 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.

Problem: Find Mathematical Model

How can we graph this scenario?

Why care?

Page 23: 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.

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.”

Page 24: 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.

B. Visual

X(t)= 4t2

Y(t)= 3t

Set up a table:

tX(t)

Y(t)

Page 25: 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.

C. Process

What do you notice?Polar Connection?

Example:

Given: x(t)=4t and y(t)= t2

How do you eliminate the parameter?

Page 26: 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.

Eliminating Parameters from Parametric Equations:

Given the following parametric equations, can you create one equivalent equation without the parameter?

Page 27: 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.

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

Goal Problems

Page 28: 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.

Solutions to Goal Problems1. D

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2. B

Page 31: 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.

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?

D. Purpose: applications that involve time as a function of two other variables

Page 32: 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.

Solutiona)

b) 3.197 seconds

c) 435.6 ft.

d) 43.44 ft at 1.55 seconds

f) Angle: -9.86° or 350.12°

Page 33: 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.

Active Practice

Recall & Reproduction Routine Non-Routine

§10.2 p590 #17-24 §10.2 p590 #25-28 §10.2 p590 #29-30

Concept Category #16 (6.3):  Parametric Equations & Graphs (Honors)