Warm-up: Solve for x. HW: Graphing Sine and Cosine Functions

SINE AND COSINE FUNCTIONS 4.5 GRAPHS OF SINE AND COSINE FUNCTIONS Objective: Graph sine and cosine functions Find the amplitude and period of the sine and cosine function Translations of sine and cosine functions Visual Aid Introduction: https://www.desmos.com/calculator/y0gfrepolx

We are interested in the graph of y = f(x) = sin x Start with a "t" chart and let's choose values from our unit circle and find the sine values. x y 1 - 1 x y = sin x

y = f(x) = sin x choose more values x y = sin x join the points y 1 x - 1 If we continue picking values for x we will start to repeat since this is periodic.

Here is the graph y = f(x) = sin x showing from -2 to 6 Here is the graph y = f(x) = sin x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 y 2 2 y = sin x x It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function)

What are the x intercepts? Where does sin x = 0? …-2, -, 0, , 2, 3, 4, 5, 6, . . . y y = sin x x Where is the function maximum? Where does sin x = 1?

Where is the function minimum? Where does sin x = -1? y y = sin x x

Knowing transformations and the graph of y = sin x, what do you suppose y = sin x + 2 looks like? The function value (or y value) is just moved up 2. y = 2 + sin x y y = sin x x

Knowing transformations and the graph of y = sin x, what do you suppose y = sin (x + /2) looks like? This is a horizontal shift by –/2 (left /2) y y = sin x x y = sin (x + /2)

Knowing transformations and the graph of y = sin x, what do you suppose y = - sin (x ) +1 looks like? This is a reflection about the x axis (shown in green) and then a vertical shift up one. y = 1 - sin (x ) y y = - sin x x y = sin x

What would the graph of y = f(x) = cos x look like? We could do a "t" chart and let's choose values from our unit circle and find the cosine values. plot these points x y 1 - 1 x y = cos x We could have used the same values as we did for sine but picked ones that gave us easy values to plot.

y = f(x) = cos x Choose more values. x y = cos x plot these points y 1 - 1 cosine will then repeat as you go another loop around the unit circle

Here is the graph y = f(x) = cos x showing from -2 to 6 Here is the graph y = f(x) = cos x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function)

Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here

What are the x intercepts? Where does cos x = 0? Where is the function maximum? Where does cos x = 1? …-4, -2, , 0, 2, 4, . . .

Where is the function minimum? Where does cos x = -1? …-3, -, , 3, . . .

Let's try y = 3 - cos (x - /4) y = - cos x y = cos x You could graph transformations of the cosine function the same way you've learned for other functions. moves right /4 moves up 3 Let's try y = 3 - cos (x - /4) y = - cos x y = cos x reflects over x axis y = 3 - cos x y = 3 - cos (x - /4)

amplitude of this graph is 2 What would happen if we multiply the function by a constant? All function values would be twice as high y = 2 sin x amplitude is here amplitude of this graph is 2 y = 2 sin x y = sin x The highest the graph goes (without a vertical shift) is called the amplitude.

For y = A cos x and y = A sin x, A  is the amplitude. What is the amplitude for the following? y = 4 cos x y = -3 sin x amplitude is 3 amplitude is 4

The last thing we want to see is what happens if we put a coefficient on the x. y = sin 2x y = sin 2x y = sin x It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes .

What do you think will happen to the graph if we put a fraction in front? y = sin 1/2 x y = sin x The period for one complete cycle is twice as long or 4

The period T = So if we look at y = sin x the  affects the period. This will be true for cosine as well. What is the period of y = cos 4x? y = cos x This means the graph will "cycle" every /2 or 4 times as often y = cos 4x

absolute value of this is the amplitude Period is 2 divided by this

Graphing Calculator!

Sneedlegrit: Graph from-2π to 2π. Remember the rules of graphing. sin(x + π/2) + 2 HW: Graphing Sine and Cosine Functions