Sinusoidal models describe quantities that rise and fall in a repeating pattern, such as ocean tides, sound waves, daylight hours, and alternating current. They are built from sine or cosine functions because these functions naturally repeat at regular intervals. In a tide model, the graph shows how water height changes over time, making it easier to predict high tides, low tides, and typical water levels.
Understanding these models helps connect algebra, graphs, and real-world periodic behavior.
A general sinusoidal model can be written as y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D. The amplitude A measures how far the graph moves above and below its midline, while D gives the midline itself. The period is found from B using period = 2π / |B|, and the horizontal shift C moves the pattern left or right.
When modeling data, students identify maximums, minimums, repeating intervals, and starting position to choose parameters that match the situation.
Key Facts
- General sine model: y = A sin(B(x - C)) + D.
- General cosine model: y = A cos(B(x - C)) + D.
- Amplitude: |A| = (maximum - minimum) / 2.
- Midline: y = D = (maximum + minimum) / 2.
- Period: T = 2π / |B|, so B = 2π / T.
- For tide models, high tide corresponds to a maximum and low tide corresponds to a minimum.
Vocabulary
- Sinusoidal function
- A function based on sine or cosine that repeats in a smooth wave pattern.
- Amplitude
- The vertical distance from the midline of a sinusoidal graph to a maximum or minimum.
- Period
- The horizontal length of one complete cycle of a repeating graph.
- Midline
- The horizontal line halfway between the maximum and minimum values of a sinusoidal graph.
- Phase shift
- The horizontal shift of a sinusoidal graph from its basic sine or cosine position.
Common Mistakes to Avoid
- Using maximum minus minimum as the amplitude is wrong because amplitude is only half the total vertical range.
- Forgetting the vertical shift gives an incorrect model because the graph may oscillate around a midline other than y = 0.
- Confusing period with frequency leads to the wrong B value because period is cycle length while frequency is cycles per unit time.
- Choosing sine or cosine without checking the starting point can shift the model incorrectly because cosine naturally starts at a maximum when A is positive.
Practice Questions
- 1 A tide has a high of 6.8 m and a low of 1.2 m. Find the amplitude and midline of a sinusoidal model.
- 2 A tide cycle repeats every 12 hours and has midline 4 m and amplitude 2.5 m. Write a cosine model that starts at high tide when t = 0.
- 3 A student models tide height with y = 3 sin(2πt / 12) + 5. Explain what the numbers 3, 12, and 5 mean in the context of the tide.