Post on 19-Jan-2016
description
Lesson 4-6Graphs of Secant and
Cosecant
2
Get out your graphing calculator…
Graph the following
y = cos x
y = sec x
What do you see??
3
2
3
y
x
2
2
2 3
2
5
4
4
xy cos
Graph of the Secant Function
2. range: (–,–1] [1, +)
3. period: 24. vertical asymptotes:
kkx 2
1. domain : all real x)(
2 kkx
cos
1sec
xx 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.
4
First graph:
• y = 2cos (2x – π) + 1
Then try:
• y = 2sec (2x – π) + 1
5
Graph
Graph the following
y = sin x
y = csc x
What do you see??
6
2
3
x
2
2
2
2
5
y
4
4
Graph of the Cosecant Function
2. range: (–,–1] [1, +) 3. period: 2
where sine is zero.
4. vertical asymptotes: kkx
1. domain : all real x kkx
sin
1csc
xx 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 function
is undefined and its graph has vertical asymptotes.
7
First graph:
• y = -3 sin (½x + π/2) – 1
Then try:
• y = -3 csc (½x + π/2) – 1
8
Key Steps in Graphing Secant and Cosecant
1. Identify the key points of your reciprocal graph (sine/cosine), note the original zeros, maximums and minimums
2. Find the new period (2π/b)3. Find the new beginning (bx - c = 0)4. Find the new end (bx - c = 2π)5. Find the new interval (new period / 4) to divide the new
reference period into 4 equal parts to create new x values for the key points
6. Adjust the y values of the key points by applying the change in height (a) and the vertical shift (d)
7. Using the original zeros, draw asymptotes, maximums become minimums, minimums become maximums…
8. Graph key points and connect the dots based upon known shape
Graphs of Tangent and Cotangent Functions
10
Tangent and Cotangent
Look at:ShapeKey pointsKey featuresTransformations
Graph
Set window
Domain: -2π to 2π
x-intervals: π/2
(leave y range)
Graph
y = tan x
11
12
y
x
2
3
2
32
2
Graph of the Tangent Function
2. range: (–, +)
3. period:
4. vertical asymptotes: nnx
2
1. domain : all real x nnx
2
Properties of y = tan x
period:
To graph y = tan x, use the identity .x
xx
cos
sintan
At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.
13
Graph
y = tan x and y = 4tan x in the same window
What do you notice?
y = tan x and y = tan 2x
What do you notice?
y = tan x and y = -tan x
What do you notice?
Graph
Set window
Domain: 0 to 2π
x-intervals: π/2
(leave y range)
Graph
y = cot x
14
15
Graph of the Cotangent Function
2. range: (–, +)
3. period: 4. vertical asymptotes:
nnx
1. domain : all real x nnx
Properties of y = cot x
y
x
2
2
2
32
3
2
xy cot
0xvertical asymptotes xx 2x
To graph y = cot x, use the identity .x
xx
sin
coscot
At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.
16
Graph Cotangent
y = cot x and y = 4cot x in the same window
What do you notice?
y = cot x and y = cot 2x
What do you notice?
y = cot x and y = -cot x
What do you notice?
y= cot x and y = -tan x
17
Key Steps in Graphing Tangent and Cotangent
Identify the key points of your basic graph1. Find the new period (π/b)2. Find the new beginning (bx - c = 0)3. Find the new end (bx - c = π)4. Find the new interval (new period / 2) to divide
the new reference period into 2 equal parts to create new x values for the key points
5. Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d)
6. Graph key points and connect the dots