Sine and cosine graphs show repeating patterns called waves, which makes them useful for modeling sound, tides, springs, circular motion, and many other periodic situations. A basic sine or cosine curve repeats forever with a regular height and length. By changing the equation, you can stretch, shrink, shift, or reflect the graph.
Learning to read these changes from the equation helps you sketch accurate graphs quickly.
Key Facts
- General sine form: y = A sin(B(x - C)) + D
- General cosine form: y = A cos(B(x - C)) + D
- Amplitude = |A|
- Period = 2π / |B|
- Phase shift = C, so the graph shifts right if C > 0 and left if C < 0
- Midline: y = D, maximum = D + |A|, minimum = D - |A|
Vocabulary
- Amplitude
- The amplitude is the distance from the midline to a maximum or minimum point of a sine or cosine graph.
- Period
- The period is the horizontal length of one complete cycle of a repeating graph.
- Phase shift
- The phase shift is the horizontal movement of a sine or cosine graph caused by the value C in x - C.
- Vertical shift
- The vertical shift is the up or down movement of the graph caused by the value D.
- Midline
- The midline is the horizontal line halfway between the maximum and minimum values of the graph.
Common Mistakes to Avoid
- Using A as the maximum value instead of the amplitude is wrong because the maximum also depends on the vertical shift D.
- Forgetting the absolute value in period = 2π / |B| is wrong because period is a positive distance, even when B is negative.
- Reading y = A sin(B(x - C)) + D as a shift left by C is wrong because x - C means the graph shifts right when C is positive.
- Graphing sine and cosine with the same starting point is wrong because basic sine starts at the midline while basic cosine starts at a maximum.
Practice Questions
- 1 For y = 3 sin(2x) - 1, find the amplitude, period, midline, maximum value, and minimum value.
- 2 For y = -2 cos(4(x - π/6)) + 5, find the amplitude, period, phase shift, vertical shift, maximum value, and minimum value.
- 3 Explain how the graph of y = sin(x) changes to become y = 2 sin(x - π/3) + 4, and describe the order of features you would mark before sketching it.