8.2 Sine and Cosine Curves

1. 8.2 Sine and Cosine Curves -transformations -period -amplitude

2. Stretching a function f(x) • Remember the function f(x)=x2. If we take that function and multiply it by some constant f(x)=cx2 then that multiple stretches the graph vertically if c is bigger than 1, and it compresses the graph if 0<c<1. • Now when we take f(x)=sin(x) and multiply it by a constant c, then the same thing should happen. Key x-axis coordinates to identify are x=0,π/2,π,3π/2,and 2π. (mainly because they are at heights of 0 and 1

3. f(x)=2sin(x) • The x-coordinates do not change, but the corresponding y values are changed by a multiple of 2. Graph it.

4. f(x)=sin(x)

5. AMPLITUDE • We refer to the height of these functions as the AMPLITUDE (how high and how low they get). • Y=2 sin(x) has an amplitude of 2. • Y= sin(x) has an amplitude of • For any function sine or cosine in the form of • Amplitude = |A| • It should then make sense that for the function y=sin(x) because A=1, then we only reach at height of 1 and a low of -1.

6. PERIOD • We have discussed that the functions y=sin(x) and y=cos(x) has a period of 2π. • That means it takes a distance of 2π on the x-axis before we see the entire function repeat itself. • This is the fundamental period of sine and cosine. • However we can change the period. We can force the sine/cosine functions to repeat themselves within a distance of π for example. • In all reality we can force the sine/cosine functions to repeat themselves within a distance of any multiple or π (whether that multiple is larger or smaller than π).

7. y=sin(2x) Note: Show with a chart on sideboard Here the x values change, but the y values do not. It is much easier to first identify what the period is and then graph using relationships.

8. y=sin(x) Note: Show with a chart on sideboard Here the x values change, but the y values do not. It is much easier to first identify what the period is and then graph using relationships.

9. Period • The period is affected by the value in front of the variable that is used to represent the angle (usually x or theta). If we refer to that constant as B, if B>1 then the period is compressed horizontally. If 0<B<1 then the period is stretched horizontally. • It is often easier to find the period prior to graphing. Then use relationships between the original sin(x) or cos(x) and the fundamental period compared to the f(x) that you are attempting to graph.

10. The period for either of these functions is • So if you think about y=cos(x) and we remember that the period is 2pi, it should make sense that this is true because in that instance B=1.

11. f(x)=sin(2x) • The x-coordinates do not change, but the corresponding y values are changed by a multiple of 2. Graph it.

12. Graph y=cos(4x) using relationships.