Explanation:
What we want to do is use the data to find the equation of the sinusoidal wave in order to plot it.
A sinusoidal wave is given by the formula
y = A* sin(kx+o)
Where A is the amplitude of the sine (distance from x axis to the highest or lowest poin the graph reaches in the y-axis), k is the scale of the horizontal axis, and o is the phase.
Let's say the y-axis is for temperature in Celsius and the x-axis is time in minutes.
We are given that the highest temperature of the liquid is 18. Hence, the amplitude is 18.
A =18 ºC
Also we know the temperature is 12 ºC at the beginning, which means
y = 12 ºC when x = 0 min.
Substitute in the formula:
12 = 18*sin(0+o)
And solve for the phase o
o = arcsin(0.666) = 0.729 rad
Then we are given that y = 18 ºC after 2 minutes (x = 2 min). So substitute in the equation to find k
18 = 18*sin(k*(2)+0.729)
k*(2)+0.729 = arcsin(1) = π/2
k = 0.420
Now substitute A and the calculated values for k and o. This is the formula of the sinusoidal wave we must plot
y = 18* sin(0.420x+0.729)
We can use the a free software to plot the formula. The final graph is attached.
The start and end of the two complete cycles are indicated by the yellow bar. One cycle is when, once the temperature is 12 degrees, it reaches 12 degrees again.