To sketch the two complete cycles of the sinusoidal function, we need to determine the amplitude, period, phase shift, and vertical shift of the function based on the given information.
The amplitude is the distance between the maximum and minimum values of the function, and is equal to (maximum value - minimum value)/2. In this case, the maximum temperature is 18°C and the minimum temperature is not given, so we'll assume it is 6°C (the average of the initial temperature of 12°C and the maximum temperature of 18°C). Therefore, the amplitude is (18 - 6)/2 = 6°C.
The period is the length of one complete cycle of the function, and is equal to the time it takes for the temperature to go through one complete cycle of heating and cooling. In this case, the time for one complete cycle is the time it takes for the temperature to go from the maximum of 18°C, to the minimum of 6°C, and back to the maximum of 18°C. From the given information, we know that the time for the first half of the cycle (heating) is 2 minutes, so the total time for one complete cycle is 2 x 2 = 4 minutes.
The phase shift is the horizontal shift of the function, and indicates how far the function is shifted to the left or right from its usual position. In this case, there is no phase shift, since the function starts at themaximum temperature of 18°C at time t = 2 minutes.
The vertical shift is the vertical displacement of the function, and indicates how far the function is shifted up or down from its usual position. In this case, the vertical shift is 6°C, since the average temperature of the liquid is 6°C higher than the minimum temperature of 6°C.
Putting all of this together, the sinusoidal function that describes the temperature of the liquid over time can be written as:
T(t) = 6 sin(πt/2) + 12
where T is the temperature of the liquid in degrees Celsius, t is the time in minutes, and the amplitude is 6, the period is 4, the phase shift is 0, and the vertical shift is 12.
To sketch two complete cycles of this function, we can use a graph with time on the x-axis and temperature on the y-axis. We can plot points for the maximum and minimum temperatures at t = 2, t = 3, t = 4, t = 5, t = 6, and t = 7 minutes, and then connect the points with a smooth curve to show the sinusoidal variation in temperature over time.
Here is a sketch of two complete cycles of the sinusoidal function:
| /\
18 | / \
| / \
|___/ \______
2 | / \ / \
15 |__/ \__/
| / \
12 |____/ \______
2 4 6
The curve starts at the maximum temperature of 18°C at t = 2 minutes, decreases to the minimum temperature of 6°C at t = 4 minutes, increases back to the maximum temperature of 18°C at t = 6 minutes, and then completes another cycle by returning to the minimum temperature of 6°C at t = 8 minutes. The curve repeats this pattern over time, showing the sinusoidal variation in temperature as the liquid is heated and cooled repeatedly during the experiment.