Track a Shifted Cosine Pattern

Supplied materials

• Paper or a note app.
• Supplied fictional rule h(t) = 5 + 3cos(π(t − 2)/6), t in hours. Use cos(0)=1, cos(π/2)=0, cos(π)=−1, cos(2π)=1; angles are radians.

Learn the idea

Replacing t with t − 2 shifts the unshifted cosine pattern two hours to the right. At t = 2 the angle is zero, so the peak occurs there. The midline five, amplitude three, and period twelve remain the same.

Work relative to the shifted start. A quarter period is three hours, so t = 5 reaches the midline, t = 8 reaches the minimum, and t = 14 reaches the next peak. The shift changes when a value occurs, not its height or repetition interval.

1. Set t − 2 = 0 to locate the first nonnegative peak at t = 2.
2. Add a quarter period to obtain t = 5 and substitute its angle.
3. Check t = 8 and t = 14 to locate a minimum and the next peak.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

For q, find the next minimum. Half the twelve-hour period after the t = 4 peak is t = 10, where q(10) = 5 + 3cos(π) = 2.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback.