Transcript 4.4 Graphing by Addition of Ordinates The period of a sum function is found by examining the period of each individual part. The period.

4.4 Graphing by Addition of
Ordinates
The period of a sum function is found by examining the
period of each individual part.
The period is the first place the periods match up.
For example, say 2 functions had periods of π and 2π.
What’s the period of the sum function?
list multiples of each until they match
π  2π
 period = 2π
2π
Something more challenging: Periods of

2


2
2

2
3



3
3
3
 
2
,
 period = π
3
Ex 1) Graph y  2cos x  sin 2 x
per = π
per = 2π
Start by graphing y  2cos x
b=1
c=0
a=2
d=0
Per =
IL:
 period = 2π
(longer period)
Check Period:
2  0  2
2
2
 2
b
1 
 2  
4 2
0
–2

2

3
2
2
Ex 1) Graph y  2cos x  sin 2 x
Now graph y  sin 2 x on same axes
b=2
c=0
a=1
d=0
Per =
IL:
Check Period:
 0 
2
1
2

2
1 
  
4 4
–1

4


2
3
4

3
2
2
–2
2  3  4
0   
 
 
4 4 4
4
4
4
4
4


0
Continue graph to
match cosine’s period
Ex 1) Graph y  2cos x  sin 2 x
Now add the 2 graphs together (y-values)
At x = 0  2 + 0 = 2
At x
At x
At x

= 4

= 2
3
= 4
3
 1 + 1.4 = 2.4
2
0+0=0
1
 –1 + –1.4 = –2.4 –1 0
At x = π  0 + –2 = –2
At x =
At x =
5
4
3
2
 1 + –1.4 = –.4
0+0=0
–2
–3

4

2
3
4

3
2
2
Let’s use Desmos to quickly graph the sum function and
explore the period.
Open Desmos. Choose
, then Trigonometry ,
and then All the Trig Functions , then Explore this Graph!
Tap into box 7 and delete boxes 4 – 7
add f (x) =
to box 2
g (x) =
to box 3
(use
)
1
1
Ex 2) Find the period of H ( x)  sin x  cos x
3
2
1
3
sin x
1
2
cos x
b
b
per 
1
3
per 
1
2
2
1
3
2
1
2
 6
 12π  18π
 12π
 4
 8π  12π
Check it with Desmos.
sin x: put
cos x: put
1
3
in front of x
1
2
in front of x
turn both graphs on
In box 4, type f (x) + g (x)
turn it on
So confusing! The period should be 12π
Turn off 2 & 3
Stretch y-axis to [–4, 4]
Pinch x-axis until you can see the graph repeat itself
At x = 0, y = 1
Does it = 1 again @ x = 12π?
YEP!!
We don’t always just graph trig + trig, sometimes we graph
trig + polynomial
Ex 3) Graph y = sin x – x using ordinate addition
(by hand from –2π to 2π)
Confirm/ Check with Desmos f (x) = sin x
g (x) = –x
f (x) = sin x
g (x) = –x
f (x) + g (x)
keep f (x) + g (x)
sin x – x
Homework
#404
Pg 211 #9, 11, 12, 21, 22, 25, 29, 31, 34, 37, 38,
39, 40, 43, 49