Precalculus
Lesson Plans - Chapter 4B – Graphing Circular Functions
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Date
1/18/2017
1/19/2017
Day
Wednesday
Thursday
1/20/2017
Friday
1/23/2017
Monday
1/24/2017
Tuesday
1/25/2017
Wednesday
1/26/2017
1/27/2017
Thursday
Friday
Objective
Sin x Graphs
Cos x Graphs
Amplitude and Period
of sin x and cos x
Phase Shifts and Vertical
Shifts of sin x and cos x
Sine and Cosine Functions:
Zeroes and Modeling
Tangent, Cotangent, Secant
and Cosecant Graphs
Review
Test
Home Enjoyment
Parish ACT practice Test 30 min. No HW
Ch. 4B HW #1 Study for Quiz #1
Quiz #1, Ch. 4B HW #2
Ch. 4B HW #3 and CTJ, Study for Quiz #2
Quiz #2, Ch. 4B HW #4
Ch. 4B HW #5, Study for Quiz #3
Quiz #3 and Homework Quiz,
Ch. 4B HW #6, Study for test
Test Ch. 4B
Make sure to answer all questions on the Homework flipchart on Moodle.
Cosine and Sine Parent Graphs
p. 299 (Sec. 4.5) #9, 10,
Study for Quiz #1: graphing basic sin x and cos x curves in radians and degrees and
first quadrant on the unit circle – radians, degrees, ordered pairs and tangent values.
#2 Cosine and Sine Functions – Amplitude and period
p. 299 (Sec. 4.5) All graphs in radians without calculator: #11-17 odd, 23, 24, 27, 29, 30, 33,
38, 57, 58, 79, 80, 82,
(A) Find the amplitude and period of each function and graph 2 periods.
(Graph by hand, check on the calculator)
(1) y = |2 sin x|, (2) y = |sin 2x|, (3) y = 2 |sin 2x |, (4) y = sin (2 |x|)
(B) Find the equation in the form y = A sin Bx or y = A cos Bx for each graph below.
(Ans:
(1) A = 1, P = p, (2) A = ½ , P = p/2, (3) A = 1, P = p/2, (4) A = 1, not periodic
(5) y = 3 cos x, (6) y = sin 2x, (7) y = 4sin
πx
, (8) y = -5 cos ½ x
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Precalculus Ch 4B Home Enjoyment
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#3 Cosine and Sine Functions – phase shifts and vertical shifts
p. 299 (Sec. 4.5) All graphs in radians without calculator: #21, 22, 25, 26, 31, 32, 35, 36, 39,
41, 49, 51, 53, 54, 63, 85, 89-92 all (Explain both true and false), 95, 96,
Study for Quiz 2: one sin graph and one cos graph with all shifts in radians
Critical Thinking Journal: (Ch 4B CTJ – 15 pts.)
(1) Mrs. D teaches the general sinusoidal equation y = A sin B(x ± C) ± D. Discuss the
meaning of each constant and how you use the constants when you graph.
(Don’t just say, “A” in the amplitude – explain in your own words what that means and how ± affects
the graph. Do the same with B, C, and D.) (12 pts.)
(2) The book uses the general sinusoidal equation y = d + a sin (bx - c). Discuss the
changes in your answers in #1. (3 pts)
#4 Cosine and Sine Functions - Zeroes and Modeling
p. 299 (Sec. 4.5) All graphs in radians without calculator: #71, 72, 73, 75, 77, 87 abcde, 88 acdef,
Work the following:
(A) Graph by hand and find all the zeroes in radians for the following problems
(1) y = 2cos x
(2) y = 1/3 sin x (3) y = sin ½x (4) y = cos 1/3 x (5) y = cos (x + π/4)
(6) y = sin (π/2)x (7) y = cos πx
(8) y = 2sin( ½x + π/4)
(9) y = 3cos(2x − π)
(B) Graph by hand and find all the zeroes in degrees.
(10) y = −3sin x
(11) y = 4cos x
(12) y = cos x + 1
(13) y = sin 4x
(C) Graph the following on your calculator and find all zeroes in radians (rounded 3 places)
(14) y = 2 sin(3x + 5)
(15) y = 3cos x + 2
(D) Graph the following on your calculator and find all zeroes in degrees
(Make sure to change to degree mode and zoom trig.)
(16) y = −3sin 7x
(17) y = 2cos(8x – 60) +1
(Ans:
(1) x = p/2 + kp, k Î J
(2) x = kp, kÎ J,
(3) x = 2kp, kÎ J,
(4) x = 3p/2 + 3kp, kÎ J,
(5) x = p/4 + kp, k Î J,
(6) x = 2k, k∈J
(7) x = ½ + k, k∈J
(8)
(10) x = 180k, k Î J,
(11) x = 90 + 180k, k Î J
(12) x = 180 + 360k, k Î J
(13) x = 45k
(14) x = 0.428 + kπ/3, kÎ J,
(15) x = 2.301+2kπ, x = 3.983 + 2kπ, kÎ J,
(16) x = 25.714k, kÎ J,
(17) x = −7.5 + 45k, kÎ J, x = 22.5 + 45k, kÎ J
3π
x =+ 2kπ , k ∈ J
2
(9)
3π kπ
x= +
, k∈J
4
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Precalculus Ch 4B Home Enjoyment
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#5 Tangent, Cotangent ,Secant, and Cosecant Graphs
p. 311 (Sec. 4.6) #5-8 all, 9, 11, 14, 15, 17, 19, 24, 27, 29, 33, 35, 37, 38, 39,
78 ab (solve without calculator)
(A) Graph the following by hand from [-2p, 2p]. Check on your calculator.
(1) y = -sin x (2) y = |sin x|
(3) y = sin |x|
(4) y = -cos x (5) y = |cos x|
(6) y = cos |x| (7) y = -tan x
(8) y = |tan x|
(9) y = tan |x|
(B) Find the following limits. If the limit does not exist state why.
(1) lim sin x (2) lim cos x (3) lim tan x (4) lim csc x (5) lim sec x (6) lim cot x
x →∞
x →∞
x→
π
2
x→
π
2
x→
π
2
x→
π
2
(Ans: (1) DNE oscillating, (2) DNE oscillating, (3) DNE unbounded, (4) 1, (5) DNE unbounded (6) 0
(C) Study for Quiz #3: Basic tangent, cotangent, secant, and cosecant graphs
#6 Test tomorrow. Know the graphs, domain, range, asymptotes, and zeroes of all six basic functions.
Be able to work all the problems on the worksheets and review. Hand in Little Blue Book.
Precalculus Ch 4B Home Enjoyment
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Precalculus Ch 4B Home Enjoyment
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Review
I.
Fill in the following information. Amplitude, period, phase shift, vertical shift
3
2
−2sin x − 5
(1) y =
II.
III.
(4)
(7)
(8)
(9)
(2) y 3cos ( 2 x + π )
=
π

=
(3) y 2 tan 4  x + 
2

State the domain and range of all six trig functions
Graph the following equations. Graph one period unless otherwise specified. The x-axis must be
marked off at the zeroes maximum and minimum points, and asymptotes. The y axis must be marked
off at the amplitude. Write the equation of the zeroes. (Don’t forget k∈J every time.) Dot
asymptotes.
π


3

y = −3 sin x (Graph 2 periods making sure you have some of the graph on both sides of the y axis)
y = cot x (Graph 2 periods making sure you have some of the graph on both sides of the y axis)
y = csc (3x) (make sure you have some of the graph on both sides of the y axis and I can tell what is the final
=
y tan  x +
y = cos x + 1, (5) y = cos (−4x) , (6)
cosecant graph)
(10)
y = | 4 sin px | (graph 2 periods of the normal y= 4 sin πx before translating, darken the real
graph making sure you have some of the graph on both sides of the y axis)
(11)
(12)
y = sin |2 x | (graph + and − periods of the normal sin graph before translating, darken the final graph)
y = cos (½ x + p) rewrite as a sine function
IV.
Find the zeroes only: (noncalculator) Circle your final answers. Remember, these are
shifted up so you can only find them algebraically and you should have two equation
answers, 1 equation answer, or the empty set.
(13)
y = cos x − ½
(14)
y=
(15)
(16)
y = 3 cos x + 3
y = −3 sin x + 7
V.
Finding equations: Determine the equation of the following graphs using exact numbers:
(17)
2
+ sin x
2
(18)
same window
VI. Calculator portion of test: Find zeroes and points of intersection on calculator from sine,
cosine and tangent graphs. Round all answers three places behind the decimal.
(1) Find all the zeroes using the first negative and the first positive zeroes to make you
equation answers. y = 4 sin (½x+3) + 1
(2) Find all the zeroes using the first positive zero to make your answer. y = 7 tan (x–2) + 1
(3) Find the first positive point of intersection of y = sin (x+2) + 13 and y = 2 cos (x−2) + 14
(Ans: (1) x = -.6505 + 4kp, x = 0.789 + 4kp, kJ, (2) x = 1.858 + kp, kJ, (3) (.301, 13.745)
,
Precalculus Ch 4B Home Enjoyment
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Precalculus Ch 4B Home Enjoyment
Little Blue Book of PreCalculus Properties
Ch. 4B – Graphing Circular Trig Functions
4B-1
4B-2
4B-3
4B-4
4B-5
4B-6
4B-7
4B-8
4B-9
4B-10
4B-11
4B-12
Graphing Terminology (define amplitude, phase shift, vertical shift, period, frequency,
zeroes, and asymptote, explain what k ∈ J means and why you use it, another symbol for
set of integers)
Sine Function (abbreviation, graph, zeros, amplitude, period, domain, range, even or odd and
justify)
Cosine Function (abbreviation, graph, zeros, amplitude, period, domain, range, even or odd
and justify)
Translations of y = A sin B(x − C) + D, y = A cos B(x − C) + D (the effects of A, B, C, & D on
period, amplitude, vertical shift & phase shift of sin & cos, how to rewrite a sine function as
a cosine function and vice versa, give ex. of translated sin graphs, what has to be done to
y = cos (3x + 6) before you can find the phase shift?)
Tangent Function (abbreviation, graph, zeros, amplitude, period, domain, range, asymptotes,
even or odd and justify)
Translations of y = A tan B(x – C) + D (the effects of A, B, C, & D on period, amplitude, vertical
shift & phase shift of tan x, give ex. of translated tan graphs)
Cotangent Function (abbreviation, graph, zeros, amplitude, period, domain, range,
asymptotes, even or odd and justify, how to graph y = cot x on calculator)
Secant Function (abbreviation, graph, explain the procedure for graphing, zeros, amplitude,
period, domain, range, asymptotes, even or odd and justify, how to graph y = sec x on calculator)
Cosecant Function (abbreviation, graph, explain the procedure for graphing, zeros,
amplitude, period, domain, range, asymptotes, even or odd and justify, how to graph y = csc
x on calculator)
Zeroes of trig functions – Steps for finding infinite number of zeroes analytically. How many
equations does the solution for zeroes of tan x need? When do you need two equations for the
solutions of zeroes of sin x and cos x? When do you use 360k and 2kπ? Find all the zeroes in
exact radians of y = 6 sin (3x-π) + 3. Find all the exact zeroes y = 7tan (3x+ 4) + 7. Show all the
work. This card will probably take a whole page.)
Zeroes from the calculator - Steps for finding infinite number of zeroes on calculator. How
many equations does the solution for zeroes of tan x need? When do you need two equations for
zeroes of sin and cos? Graph in radian mode and state all the zeroes for y = 5sin (4x - 2) + 3)
Modeling with Sinusoidal Regressions – (What information is needed to find the sine or
cosine equation analytically? Give a real world example with a scatter plot. Steps for finding
a sinusoidal regression equation in your calculator.)
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