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Trigonometric Functions – Lesson 3
REVIEW:Graph Sine and Cosine
FunctionsWith amplitude and period
changes.INVESTIGATE”
VERTICAL SHIFTObjective: To graph sine and
cosine functions with amplitude, period changes
and vertical shift!
What do trig functions model in real life?
• Sound waves• Ferris Wheel• Music frequencies• EKG’s• Just as we can create linear, exponential
and quadratic models to represent real life data, we can also use regression to determine whether or not a trig function would be a good model to represent the data!
Graphing: What do we know – starting point?
f(x) = sin x and f(x) = cos x
Range & Intercepts:f(x) = sin x and f(x) = cos x what is the shift between sin and cos?
f(x) = sin x & two important ideas
Period
Am
plitudePeriod
Period means how many degrees in one cycle.
Amplitude means the distance from the centre to the
maximum or minimum, OR (max + min) ÷ 2A
mplitude
f(x) = sin x
Period
Period = 360º
Amplitude = 1
How does “b” impact the graph?f(x) = sin x & f(x) = sin 2x
Period = 180º
b = 2What does it do?
f(x) = sin x & f(x) = sin 3x
So b changes the period = 360º ÷ b or
If _____ ____1 it’s hard to get out of the water! If _____ ___ 1 it’s easy to get out without getting slammed by a wave!
Period
= 120º
How does “A” impact the graph?f(x) = sin x & f(x) = -1 sin xIs the y intercept the same?
What changes?
Amplitude = 1
f(x) = sin x & f(x) = -3 sin x
Amplitude = 3
f(x) = sin x & f(x) = A sin x
The A gives the amplitude of the function.
A negative value means the graph goes down – up, not up – down.
A = 4
A = -3
Amplititude = “a”If ______ ____ 1 you get a taller
wave! (Think: Hawaii Waves!)
If ________ ___ 1 you get CT shore waves!
Now we will investigate how k impacts the graph!
f(x) = a sin bx + k
Any conjectures about “K”? Where have we seen “k” before?
What did the “K” do in this function?
f(x) = sin x & f(x) = sin x + 3
f(x) = sin x + 3 & f(x) = sin x – 2
So “k” shifts the curve up and down. We call this vertical shift or vertical displacement.
f(x) = asin bx + k
a = amplitude
Note: It is exactly the same for sine and cosine.
The difference is the where it crosses the y-axis.
b = 360º ÷ period
k = vertical shift
What is the equation of this function?
Amplitude = 2
Period = 120º
Vertical shift = -1
f(x) = 2 sin 3x – 1
so, A = 2
so, B = 3
so, k = -1
What is the equation of this function?
Amplitude = 4,
going down-up
Period = 720º
Vertical shift = 1
f(x) = -4 sin ½x + 1
so, A = -4
so, B = 0.5
so, k = 1
What is the equation of this function?
Amplitude = 2.5
Period = 240º
Vertical shift = 2
f(x) = 2.5 sin 1.5x + 2
so, A = 2.5
so, B = 1.5
so, k = 2
Sketch the graph of y = 2 sinx - 4
A = ________ b = __________ k = _______
Period = _______________
5 critical points:Range:
Sketch the graph of y = sin2x - 4
A = ________ b = __________ k = _______
Period = _______________
5 critical points:
Range:
Put it altogether! Sketch the graph of
a = ________ b = __________ k = _______
Period = _______________
5 critical points
Range
Graph. Then check your graph on your gc!
1. Y = -2cosx + 4 2. y = .5sin2x - 2