Learning goal and prerequisites
Construct a sinusoid from extrema and timing, then interpret changes in context. Before starting, review sinusoidal models.
Method and conditions
Use half the extreme difference for amplitude and their mean for the midline. Consecutive peak spacing gives the period. A cosine model can be anchored at a known maximum: M+A cos(2π(t−h)/T).
Worked example: show the decisions
A buoy reaches height 9 at consecutive maxima t=2 and t=10; its minimum is 3. Amplitude is 3, midline 6 and period 8, so H(t)=6+3cos(π(t−2)/4). It reaches the midline descending at t=4.
Contrast: why a plausible shortcut fails
A time shift of two is inside the expression t−2; adding two outside changes height. A peak-to-trough interval is half a period, whereas consecutive peaks are one full period apart.
Transfer: try before revealing the answer
A signal has consecutive maxima of12 at t=1 and t=7, minimum 4. Write a model and find the average rate from t=1 to t=4.
Transfer solution and check
H(t)=8+4cos(π(t−1)/3). At t=4 the height is 4, so the average rate is (4−12)/(4−1)=−8/3 height units per time unit. Its first descending midline crossing is one quarter-period after the peak: t=1+6/4=5/2.
H(t)=7+3cos(π(t−3)/5). What is the first nonnegative time t at which H reaches its maximum? Here 3<10.
- A cosine maximum has inner angle zero modulo 2π; t=3 is first because the preceding maximum is at -7<0.
- Using these quantities: 3 = 3.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Compare the other answer choices
Use half the period as the phase shift.
Report the period instead of the time of the first maximum.
Reverse the sign of the horizontal shift.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance