LESSON 14 – FUNDAMENTAL THEOREM of ALGEBRA PreCalculus - Santowski.
Precalculus Lesson
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Transcript of Precalculus Lesson
PRECALCULUSLESSON
Check: p. 326 [Evens] # 12-18, 32-36, 50-54
4.6
WARM-UP
1. ( ) 4sin2. ( ) sin(2 ) 2
3. ( ) 3cos2
f x xg x x
h x x
Graph 2 full periods.
ANSWERS
On other sheet
OBJECTIVE:
Graph sine cosine and tangent functions, and their reciprocals.
Graphs of Trigonometric Functions
4.6 Digital Lesson
Tangent, cotangent, secant, and cosecant
y
x
23
23
2
2
Graph of the Tangent Function
2. range: (–, +) 3. period: 4. vertical asymptotes:
kkx 2
1. domain : all real x kkx
2
Properties of y = tan x
period:
To graph y = tan x, use the identity .xxx
cossintan
At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.
2. Find consecutive vertical asymptotes by solving for x:
4. Sketch one branch and repeat.
Example: Find the period and asymptotes and sketch the graph of xy 2tan
31
22 ,
22
xx
4 ,
4
xxVertical asymptotes:
)2
,0( 3. Plot several points in
1. Period of y = tan x is.
2 . is 2tan of Period xy
x
xy 2tan31
8
31
0
08
31
83
31
y
x2
83
4
x4
x
31,
8
31,
8
31,
83
Graph of the Cotangent Function
2. range: (–, +) 3. period: 4. vertical asymptotes:
kkx
1. domain : all real x kkx
Properties of y = cot x y
x
2
2
23
23
2
xy cot
0xvertical asymptotes xx 2x
To graph y = cot x, use the identity .xxx
sincoscot
At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.
Example 1: Graph a) tan2xy
Consecutive Asymptotes 2 2
bx c
Period: difference between consecutive asymptotes
Add or subtract the Period to Consecutive asymptotes to
complete 2 periods.
Graph 2 full periods (cycles)• Asymptotes and intercepts.
Example 1:Graph b)
0 bx c
Period: difference between consecutive asymptotes
Add or subtract the Period to Consecutive asymptotes to
complete 2 periods.
cot2
y x
Consecutive Asymptotes
Graph 2 full periods (cycles)• Asymptotes and intercepts.
23
y
x
2
2
2 32
5
4
4
xy cos
Graph of the Secant Function
2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes:
kkx 2
1. domain : all real x)(
2 kkx
cos
1secx
x The graph y = sec x, use the identity .
Properties of y = sec x
xy sec
At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.
23
x
2
2
22
5
y4
4
Graph of the Cosecant Function
2. range: (–,–1] [1, +) 3. period:
where sine is zero.
4. vertical asymptotes: kkx
1. domain : all real x kkx
sin
1cscx
x To graph y = csc x, use the identity .
Properties of y = csc x xy csc
xy sin
At values of x for which sin x = 0, the cosecant functionis undefined and its graph has vertical asymptotes.
Example 2 graph a)
:amplitude a
endpoints for one period: 0 bx-c 2
2:periodb
x
y
:4
periodInterval
2csc4
y x
Table for 2sin4
y x
Graph 2 full periods (cycles)x
y
Example 2 graph b)
:amplitude a
endpoints for one period: 0 bx-c 2
2:periodb
x
y
:4
periodInterval
3sec2y x
Table for 3cos2y x
Graph 2 full periods (cycles)x
y
Classwork: p. 337 # 17, 33, 35
Homework(4.6)(4.6) p. 337 # 16, 18, 22, 34(4.5) p. 327 # 40, 42, 56, 60
Assignments Unit Circle QuizTomorrow
No Calculator
Find the amplitude & period for each.1. f(x) = 2cos 4x2. G(x) = -5sin 2x3. H(x) = tan x4. P(x) = 3 cot 2x
CLOSURE