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Read the observations and units, estimate the midline, amplitude and period, fit an appropriate sinusoidal model, and test its residuals and predictions.
Remember Report predictions at a precision supported by the displayed data and the fitted model.
Worked example 1
01 / The complete original signed laboratory sensor data use signed signal and seconds.
Complete original simulated dataset: 8 observations. No omitted or interpolated observations.
seconds
signed signal
0
2.56
2
1.93
4
−0.8
6
−2.9
8
−2.53
10
−0.3
12
1.96
14
2.32
Construct a scatterplot of every pair with labeled scales. Compare the full pattern with linear, exponential, saturating and quadratic trends. Identify a rising and a falling interval using calculated secant slopes; distinguish repeated cycles from a single U-shaped turn. Do not calibrate parameters or run a regression in this task. Also classify each of these five supplied plots and justify the distinction between one U-turn and repetition.Straight trendMultiplicative trendLeveling trendU-shaped trendRepeating trend Show the requested calculations and interpretation. Written reasoning and graphs require teacher review; selecting a summary does not verify them. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / Every original observation; no connecting line
03 / The full input horizon is [0,14]; the output range of the observations is [−2.9,2.56] in signed signal. All 8 observations are plotted.
04 / On [10,12], the secant slope is 21.96+0.3=1.13. On [2,4], it is 2−0.8−1.93=−1.365. Thus the record contains both genuine increases and decreases.
05 / Count each consecutive run of tied readings as one sampled extremum when the neighboring readings are lower ( peak ) or higher ( trough ). Listed run endpoints are sample times, not a claim that the underlying signal is constant between them. Use the first sample in each run as its representative time. Interior sampled peak times or runs: none inside the sampled horizon. Interior sampled trough times or runs: 6. There are not two interior sampled peaks, so this record alone does not directly measure a repeated peak-to-peak period.
06 / The endpoint change is 2.32−2.56=−0.24, but the full excursion is 2.56+2.9=5.46. An endpoint secant hides the interior reversals and does not justify a line.
07 / A sinusoidal trial is reasonable because the record falls and rises around a bounded central level rather than sustaining one direction. A line has constant slope; a positive-base exponential with fixed nonzero coefficient is monotone; a usual saturating trend approaches a level without repeated reversals. A single U-turn is compatible with a quadratic and does not alone establish a repeating cycle. Here the full sampled turning pattern supports testing a sinusoid, not declaring it uniquely proved.
08 / Finite, noisy samples do not prove exact periodicity, identify a unique unrestricted frequency, or establish future repeats. A translated origin changes horizontal labels, not the order or spacing of observations. Every endpoint remains part of the evidence; choosing a shorter favorable window would discard relevant data.
09 / There are not two interior sampled trough runs, so a repeated trough-to-trough spacing cannot be measured from this record alone.
13 / Quadratic: one turning point and symmetric rise, not a repeating cycle. U-shaped trend
14 / Sinusoidal: repeated peaks and troughs at similar spacings. Repeating trend
Worked example 2 · Read full datasets and choose a model family
The complete original slow industrial sensor data use output units and seconds.
Complete original simulated dataset: 15 observations. No omitted or interpolated observations.
seconds
output units
0
276.1
25
279.3
50
264.6
75
243.8
100
222.3
125
198.8
150
183.2
175
182.4
200
195.5
225
220.8
250
247.1
275
264.2
300
275.1
325
276.4
350
260.9
Construct a scatterplot of every pair with labeled scales. An analyst chooses a straight line using only the first and last observations. Calculate the endpoint change and full observed range; use interior rising and falling secant slopes to assess that choice. Do not calibrate parameters or run a regression in this task. Show the requested calculations and interpretation. Written reasoning and graphs require teacher review; selecting a summary does not verify them. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Every original observation; no connecting line The full input horizon is [0,350]; the output range of the observations is [182.4,279.3] in output units. All 15 observations are plotted. On [225,250], the secant slope is 25247.1−220.8=1.052. On [100,125], it is 25198.8−222.3=−0.94. Thus the record contains both genuine increases and decreases. Count each consecutive run of tied readings as one sampled extremum when the neighboring readings are lower (peak) or higher (trough). Listed run endpoints are sample times, not a claim that the underlying signal is constant between them. Use the first sample in each run as its representative time. Interior sampled peak times or runs: 25, 325. Interior sampled trough times or runs: 175. Gaps between the declared representative peak times are 300. These are estimates influenced by sampling and noise. The endpoint change is 260.9−276.1=−15.2, but the full excursion is 279.3−182.4=96.9. An endpoint secant hides the interior reversals and does not justify a line. A sinusoidal trial is reasonable because the record falls and rises around a bounded central level rather than sustaining one direction. A line has constant slope; a positive-base exponential with fixed nonzero coefficient is monotone; a usual saturating trend approaches a level without repeated reversals. A single U-turn is compatible with a quadratic and does not alone establish a repeating cycle. Here the full sampled turning pattern supports testing a sinusoid, not declaring it uniquely proved. Finite, noisy samples do not prove exact periodicity, identify a unique unrestricted frequency, or establish future repeats. A translated origin changes horizontal labels, not the order or spacing of observations. Every endpoint remains part of the evidence; choosing a shorter favorable window would discard relevant data. There are not two interior sampled trough runs, so a repeated trough-to-trough spacing cannot be measured from this record alone. Written-work review:
Plot all observed pairs with their actual time labels and units.
Use the actual requested numerical evidence to compare model families; distinguish a modeling suggestion from proof.
Accept equivalent forms at the stated precision. Written reasoning and graphs require teacher review.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.