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.
Understanding Math: Graphing Sine and Cosine
A reliable graph begins with the midline, not with the curve. The vertical shift tells you where the wave is balanced. Mark that horizontal line first.
Then use the amplitude to locate the highest and lowest possible values. These are equally far above and below the midline. This step prevents a common error where students measure the amplitude from the horizontal axis instead of from the midline.
If a coefficient in front of sine or cosine is negative, the distance from the midline does not change. The graph is reflected across its midline.
For sine, the usual first movement from its midline goes downward instead of upward. For cosine, the usual starting high point becomes a low point.
The period controls the horizontal spacing of one complete cycle. A complete cycle means the graph returns to the same height, moving in the same direction. After finding the period, divide it into four equal parts.
These quarter-period marks give five important points for one cycle. At each mark, sine or cosine reaches a midline crossing, a peak, or a trough. Plotting these landmarks is safer than trying to draw the whole curve from a table of random values.
Join them with a smooth curve. The graph should not have sharp corners because the height changes continuously.
Horizontal shifts can be confusing because the sign inside parentheses seems reversed. A form with x minus a positive number moves right. A form with x plus a positive number moves left.
This happens because the input must make the expression inside the parentheses equal to zero before the basic pattern begins. Find that starting location before plotting the quarter-period points. For cosine, a positive amplitude begins at a maximum on the shifted starting line.
For sine, a positive amplitude begins on the midline and rises. These starting behaviors help students tell the functions apart even when both have the same amplitude and period.
Angles matter when graphing trigonometric functions. Most textbook formulas use radians, not degrees. One full turn around a circle is two pi radians, which is the natural length of a basic sine or cosine cycle.
A calculator set to degree mode gives different values from one set to radian mode, so check the mode before making a table. In real data, the horizontal variable may represent time, distance, or angle. The vertical variable may represent displacement, voltage, temperature, or another measured quantity.
A model is useful only when its period and units fit the situation. Real measurements may wobble around the ideal curve, so look for the midline, peak spacing, and overall repeating pattern rather than expecting every data point to land exactly on the graph.
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.