1. 4.4 Graphing by Addition of Ordinates

2. 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
Something more challenging: Periods of
 2π
 period = 2π
 period = π

3. Ex 1) Graph b = 1 c = 0 a = 2 d Per = IL: per = 2π per = π
 period = 2π
Start by graphing
(longer period)
Check Period: b c a d
= 1
= 0
= 2 2
Per =
IL: –2

4. Ex 1) Graph b = 2 c = 0 a = 1 d Per = IL: Now graph on same axes
Check Period: b c a d
= 2
= 0
= 1 2 1
Per =
IL: –1 –2
Continue graph to match cosine’s period

5. Ex 1) Graph Now add the 2 graphs together (y-values)
At x = 0  = 2
At x =  = 2.4
At x =  = 0
At x =  –1 + –1.4 = –2.4
At x = π  0 + –2 = –2
At x =  1 + –1.4 = –.4 3 2 1 –1 –2 –3

6. 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 (use )
g (x) = to box 3

7. Ex 2) Find the period of
Check it with Desmos.
sin x: put in front of x
cos x: put in front of x
 12π
 18π
 12π
 8π
 12π
turn both graphs on

8. In box 4, type f (x) + g (x)
So confusing! The period should be 12π
Turn off & 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 x = 12π?
YEP!!
turn it on

9. Ex 3) Graph y = sin x – x using ordinate addition
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
Try On
Your Own
keep f (x) + g (x)
f (x) = sin x
g (x) = –x
f (x) + g (x)
sin x – x

10. Homework
# Pg 211 #9, 11, 12, 21, 22, 25, 29, 31, 34, 37, 38, 39, 40, 43, 49