Trigonometric Functions – Lesson 3 REVIEW: Graph Sine and Cosine Functions With amplitude and period changes. INVESTIGATE” VERTICAL SHIFT Objective: 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 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 Amplitude Amplitude Period Period means how many degrees in one cycle. Amplitude means the distance from the centre to the maximum or minimum, OR (max + min) ÷ 2

f(x) = sin x Period = 360º Amplitude = 1 Period

How does “b” impact the graph?f(x) = sin x & f(x) = sin 2x What does it do? Period = 180º

f(x) = sin x & f(x) = sin 3x So b changes the period = 360º ÷ b or = 120º 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!

How does “A” impact the graph How does “A” impact the graph?f(x) = sin x & f(x) = -1 sin x Is 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.

Amplititude = “a” If ______ ____ 1 you get a taller wave 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 b = 360º ÷ period k = vertical shift Note: It is exactly the same for sine and cosine. The difference is the where it crosses the y-axis.

What is the equation of this function? Amplitude = 2 so, A = 2 so, B = 3 Period = 120º so, k = -1 Vertical shift = -1 f(x) = 2 sin 3x – 1

What is the equation of this function? Amplitude = 4, going down-up so, A = -4 Period = 720º so, B = 0.5 Vertical shift = 1 so, k = 1 f(x) = -4 sin ½x + 1

What is the equation of this function? Amplitude = 2.5 so, A = 2.5 Period = 240º so, B = 1.5 Vertical shift = 2 so, k = 2 f(x) = 2.5 sin 1.5x + 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