Graphing Sine and Cosine
Section 4.5
Objectives
Students will be able to…
•
•
•
•
•
•
Use the unit circle to generate the parent graphs of sine and cosine
Recognize the sine and cosine parent graphs by their shape
Graph the sine and cosine functions using transformations
Determine all characteristics given the equation (amplitude, phase shift, vertical shift)
Determine starting point and ending point of graph (5 points labeled on x-axis – 1 cycle)
Write the equation given the characteristics of the trig function or the graph
Discovery
• Today, you are going to discover what the graphs of sine and
cosine look like.
• Complete the table of values given to you using your Unit Circle.
• Plot those points on a graph and connect the dots.
• Enter your values into the stat plot on your calculator and plot the
scatterplot.
• Go to stat, calc, sinreg.
• Does your graph match the one on the calculator?
This is what you SHOULD have gotten!
•
1.
2.
Graph
y = sin x
y = cos x
Angle
(degrees)
Value
Cos
Angle
(degrees)
0
0
30
30
45
45
60
60
90
90
120
120
135
135
150
150
180
180
210
210
225
225
240
240
270
270
300
300
315
315
330
330
360
360
390
390
405
405
420
420
450
450
480
480
495
495
510
510
540
540
Value
Sin
Changing the Graph
• Just like any other parent graph, we can shift and change this
graph in any way that we want.
• First though, what is the domain and range of our parent
functions?
• What is the period of the graphs?
• What is the amplitude?
𝒂 ∗ 𝒇(𝒃(𝒙 − 𝒄)) + 𝒅
• Again, using the same equation as before to shift the graph.
• Let’s see what happens when we change these letters.
Graphs!
What Does Each Parameter Do?
• General form
y  a sin(bx  c)  d
• “a” stands for _____________
•
Makes graph ___________ or _____________.
• “b” determines the ______________
•
•
b>1 period _____________
0<b<1 period _____________
• “c” moves the graph __________ or __________.
•
•
C<0 graph moves _____________.
C>0 graph moves ______________.
• “d” moves graph ______ or _________.
•
•
d>0 moves graph _________.
d<0 moves graph __________.
To Graph…
Procedures:
1.
2.
3.
4.
Find the …
Graph the …
Use amplitude to graph the …
Adjust for …
a)
5.
And then for …
Plot the…
a)
Then …
PRACTICE!!
1. 𝑦 = 2 sin 𝑥
3. 𝑦 = sin(2𝑥)
5. 𝑦 = sin 𝑥 − 90°
2. 𝑦 = 2 cos 𝑥 + 3
4. 𝑦 = cos
1
𝑥
2
6. 𝑦 = cos 𝑥
+4
𝜋
+
4
When would I use this?
x(month)
y(temperature)
Jan
21
in degrees Fahrenheit in Albany, NY,
Mar
34
for six months are given in the table.
May
58
Jul
72
Sept
61
Nov
40
• The normal monthly temperature
• Create a scatterplot of the data.
How?
Find a trigonometric model that fits the data.
Steps to follow:
•
•
•
•
•
•
Enter your data in LIST
Check your WINDOW settings to make sure your values will be seen in the
viewing window
Turn on Plot1
STAT  CALC  SinReg
•
This will give you the values for the sine curve created by the data
To graph the sine function created …
•
•
•
Y=
VARS
Statistics  EQ  RegEq
GRAPH
•
What is the period of the model?
•
Find the normal temperature in December.
•
A painting company will accept exterior jobs only when the normal
temperature is 64 degrees or higher. During what month will this
company accept exterior jobs?
Closure
On Post – It Note:
Identify the amplitude, period and 5 critical points
1
𝜋
𝑦 = sin 𝑥 −
2
3
Homework
• Worksheet 4.5 (identify amplitude, period and 5
critical points)
• No Graphing (yet!)