Precalculus Notes: Unit 9 – Vectors, Parametrics, Polars, & Complex Numbers
Date:______9.5 Parametric Equations
Syllabus Objective: 1.10 – The student will solve problems using parametric equations.
Parametric Curve: the set of all points , where and are continuous functions of t on an interval I (called the parameter interval)
Parameter: the variable t
Parametric Equations: and
Orientation: the directions that results from plotting the points as the values of t increase
Graphing Parametric Equations
Ex1: Graph
Note: Choose appropriate values for t first. Then substitute in and find x and y.
t / x / y0
1
−1
2
Make a table:
Plot points:
Eliminating the Parameter
- Solve one of the equations for t. (Or if a trig function, isolate the trig function.)
- Substitute for t in the other equation. (Or use an identity if a trig function.)
Ex2: Write the parametric equation as a function of y in terms of x.
a)
Solve for t in the x-equation (easier to solve for):
Substitute into the y-equation:
b)
Solve for t in the x-equation (easier to solve for):
Substitute into the y-equation:
Ex3: Write the parametric equation as a function of y and graph.
Note: A parametric equation can be written in terms of θ instead of t.
The x-equation is already solved for the trig function:
Substitute into the y-equation:
Graph:
Using a Trig Identity
Ex4: Eliminate the parameter.
Solving for a trig function won’t help, so we need to use the identity .
Square both equations:
Add the equations:
Trig identity:
Note: The graph is a circle. The parameter interval lets us know that it would go around 1 time.
Writing a Parameterization
Ex5: Find the parameterization of the line segment through the points and .
Sketch a graph:
; is a scalar multiple of , so
These equations define the LINE.
Find the parameter interval for the line segment: We want .
: So,
Solution:
Simulating Horizontal Motion
Ex6: A dog is running on a horizontal path with the coordinates of his position (in meters) given by where . Use parametric equations and a graphing calculator to simulate the dog’s motion.
Choose any horizontal line to simulate the motion: We will choose .
Parametric Equations:
Graph (Calculator must be in Parametric mode):
Note: To see the motion, change the type of line to a “bubble”. If you would like the bubble to move slower, make the Tstep smaller.
Parametric Equations for Projectile Motion
distance: height:
Note: On Earth, or .
Ex7: A golf ball is hit at 150 ft/sec at a 30° angle to the horizontal.
a) When does it reach its maximum height?
Height: (because a golf ball is hit from the ground)
Simplify:
Maximum height is at vertex:
b) How far does it go before it hits the ground?
Hits the ground when :
Note: We could have doubled the time it took for the ball to reach its highest point!
Distance:
c) Does the ball hit a 6 ft tall golfer, standing directly in the path of the ball 580 feet away?
Find the time it takes for the ball to be 580 ft away:
Find the height of the ball at this time:
No – the ball misses him by about ______feet.
Application: Ferris Wheel
Ex8: Ryan is on a Ferris wheel, alone, still looking for someone to share it with, of radius 20 ft that turns counterclockwise at a rate of one revolution every 24 sec. The lowest point of the Ferris wheel (6 o’clock) is 10 ft above ground level at the point (0, 10) on a rectangular coordinate system. Find the parametric equations for the position of Ryan as a function of time t (in seconds) if the Ferris wheel starts (t = 0) with Ryan at the point (20, 30).
Time to complete one revolution = 24 sec:
When :
You Try: A baseball is hit at 3 ft above the ground with an initial speed of 160 ft/sec at an angle of 17° with the horizontal. Will the ball clear a 20-ft wall that is 400 ft away?
QOD: How would you write a parametrization for a semicircle?
Reflection:
Date:______9.5 Parametric Equations Continued
Syllabus Objectives: 1.10 – The student will solve problems using parametric equations. 1.5 – The student will find the inverse of a given function. 1.6 – The student will compare the domain and range of a given function with those of its inverse.
Parametric Equations: a pair of continuous functions that define the x and y coordinates of points in a plane in terms of a third variable, t, called the parameter.
Ex1: Find determined by the parameters for the function defined by the equations . Find a direct relationship between x & y and indicate if the relation is a function. Then graph the curve.
Create a table for t, x, & y.
t / /−2
−1
0
1
2
3
Solve for t and substitute to find a direct relationship between x and y.
Graph by plotting the points in the table.
Application – Parametric Equations
Ex2: A stuntwoman drives a car off a 50 m cliff at 25 m/s. The path of the car is modeled by the equations . How long does it take to hit the ground and how far from the base of the cliff is the impact?
The car hits the ground when .
The distance from the base of the cliff is x.
Parametric Equations on the Graphing Calculator
Ex3: Graph the situation in the previous example on the calculator.
Change to Parametric Mode.
Type each equation into the Y= menu and choose an appropriate window.
This is the path of the car. Select the best viewing window for the graph.
Reflection:
Date:______9.6 Polar Coordinates
Syllabus Objectives: 3.3 – The student will differentiate between polar and Cartesian (rectangular) coordinates. 6.2 – The student will transform functions between Cartesian and polar form. 6.4 – The student will solve real-world application problems using polar coordinates.
Polar Coordinate: ; r: the directed distance from the pole (origin); θ: the directed angle from the polar axis (x-axis)
Plotting Points on a Polar Graph
Ex1: Plot the points .
Point A: Start at the polar axis and go counter-clockwise (270°). Place the point 3 units from the pole (origin). Point B: Start at the polar axis and go clockwise 240°. Place the point 8 units from the pole. (Note: Each radius drawn in the grid is 15°.) Point C: Start by going counter-clockwise (150°) from the polar axis. Place a point 2 units from the pole. Because , you must place the point on the opposite side of the pole.
Writing the Polar Coordinates of a Point
Ex2: Find four different polar coordinates of P.
Note: There are infinitely many correct answers!
Polar Conversions
Polar to Rectangular:
Rectangular to Polar:
Converting from Polar to Rectangular Coordinates
Ex3: Convert to rectangular coordinates.
a)
b)
Converting from Rectangular to Polar Coordinates (Note: Be careful with the quadrant!)
Ex4: Convert to polar coordinates.
a)
______is in Quadrant II, so the polar coordinates are______.
b)
This point is on the negative y-axis, so we know ______.
Note: There are other possible answers to these!
Converting from Polar to Rectangular Equations
Ex5: Convert the equations and sketch the graph.
a)
b)
c)
Graphing in Polar Coordinates on the Calculator
We will check our graphs above. Calculator must be in Polar mode.
a) c)
Note: We cannot check the graph of b) on the calculator, but the line represents the angle for all values of r.
Converting from Rectangular to Polar Equations
Ex6: Convert the equations.
a)
:
b)
c)
Expand:
Substitute:
So or . But is a single point. So ______
Application: Finding Distance
Ex7: The location of two ships from the shore patrol station, given in polar coordinates, are . Find the distance between the ships.
Sketch a diagram:
Note: The angle between the ships (from the patrol station) is .
Using the Law of Cosines:
You Try: Convert the coordinates. Polar: ; Rectangular:
QOD: How could you write an expression for all of the possible polar coordinates of a point?
Reflection:
Date:______9.7 Polar Graphs
Syllabus Objectives: 6.3 – The student will sketch the graph of a polar function and analyze it.
3 Main Graphs: Heart or loop (limacon), Roses (daisies), Fake Rose (lemniscates)
Equation:
Loop “less than loop”
Heart “Great Heart”
Ex.1: Transform to Cartesian
I. Heart or Loop Graphing Polar Curves
Ex. 2 Graph
Ex. 3 Graph
Ex4: Graph and find the domain, range, symmetry, and maximum r-value.
a)
Domain: Range: Max r-value:
Symmetry: Substitute . Symmetric about the
(This curve is called a limaçon.)
II. Roses (daisies)
If “n” is odd. n= # of petals
If “n” is even. 2n = # of petals
Ex. 5a.)
b)
θr
Domain: Range: Max r-value:
Symmetry: Substitute . Symmetric about
This curve is called a rose.
III. Fake Rose (lemniscates)
Ex. 6 a.)
b)
Domain: Range: Max r-value:
Symmetry:
Substitute . Symmetric about
Substitute . Symmetric about origin
This curve is called a lemniscate.
Substitute . Symmetric about y-axis Tests for
Symmetry of Polar Curves
- Symmetry about x-axis: is equivalent to
- Symmetry about y-axis: is equivalent to
- Symmetry about origin: is equivalent to
Classifications of Polar Curves
· Limaçon Curves: and
· Rose Curves: and
Petals: odd = n and even = 2n
· Lemniscate Curves: and
You Try: Use your graphing calculator to explore variations of . Describe the effects of changing the window, the θ-step, a, n, and changing to .
QOD: Are all polar curves bounded? Explain.
Reflection:
Page 1 of 14 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 9