Modeling a periodic context
A cold winter day
- Unit 3
- 6 points
- About 18 minutes
A repeating real-world situation is modeled by a sinusoidal function. You find coordinates of key points on its graph, write the function's equation, and describe how the function, its concavity and its rate of change behave on an interval. No calculator. On the exam: Question 3 of 4. Part B of the free-response section: 2 questions in 35 minutes, no calculator allowed.
The question and its sources
No calculator is allowed for this question. Angle measures are in radians. Give exact values (fractions and multiples of π are fine). On the exam this question shows a graph; here the graph is described in words.
The situation
On a winter day in a cold city, the air temperature rises and falls in a daily cycle. The temperature is lowest, −8 degrees Celsius (°C), at time t = 4 hours after midnight. It is next at its highest, 4 °C, at t = 16 hours. Assume the temperature keeps rising and falling in this same pattern every 24 hours.
The sinusoidal function h models the air temperature, in °C, as a function of time t, in hours after midnight. A positive value of h(t) means the temperature is above 0 °C; a negative value means it is below 0 °C.
Source: Hypothetical scenario
Graph of h (described)
The graph of h and its dashed midline are drawn for two full cycles. No axes or scale are shown.
Five points, F, G, J, K and P, are labeled on the graph in order from left to right, one after another: F is where the graph crosses its midline while rising; G is the next maximum; J is the next point where the graph crosses its midline while falling; K is the next minimum; P is the next point where the graph crosses its midline while rising.
Source: Description of a hypothetical graph
Suggested time: 18 minutes
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Part (a)
2 pointsDetermine possible coordinates (t, h(t)) for the five points F, G, J, K and P.
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Part (b)
2 pointsThe function h can be written in the form h(t) = a sin(b(t + c)) + d. Find values of the constants a, b, c and d.
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Part (c)
2 pointsRefer to the graph of h described above. The t-coordinate of K is t₁, and the t-coordinate of P is t₂. (i) On the interval (t₁, t₂), which of the following is true about h? (a) h is positive and increasing. (b) h is positive and decreasing. (c) h is negative and increasing. (d) h is negative and decreasing. (ii) On the interval (t₁, t₂), describe the concavity of the graph of h and determine whether the rate of change of h is increasing or decreasing.
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