You are given a table of average monthly temperatures and a...

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the internet through the following website created by Robert Hoare: http://www.worldclimate.com/. (TABLE CANNOT COPY)

Key Concepts

Phase Shift

The phase shift determines the horizontal displacement of the sine function along the time axis. It adjusts the starting point of the function’s cycle to match the timing of known data patterns, ensuring that the peaks and troughs align with the observed high and low temperature periods.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. In the context of modeling temperature data, periodic functions are essential because they capture the seasonal variations observed across different months of the year.

Sine Function

The sine function is a basic periodic function that is especially useful in modeling phenomena with regular, cyclic behavior. Its smooth, wave-like curve makes it ideal for approximating patterns in data that display oscillatory trends, such as monthly temperature variations.

Amplitude

Amplitude represents the height of the peaks and the depths of the troughs in a sine wave, indicating the maximum deviation of the function's values from its midline. In temperature modeling, the amplitude quantifies the extent of variation between the hottest and coolest months.

Period

The period of a sine function is the length of the interval over which the function completes one full cycle. This concept is crucial when aligning the function with real-world cyclic phenomena, such as the annual cycle of seasonal temperature changes.

Vertical Shift

The vertical shift moves the sine function up or down along the y-axis, setting a new baseline for the oscillation. For temperature modeling, the vertical shift represents the average temperature over the period, around which the seasonal variations occur.

Transcript

00:01 We have average temperature data for one full year in phoenix, arizona, and we plotted these points over a period of two years.

00:10 So using this graph that we plotted these points onto, it appears that we have something of a periodic negative cosine curve going on here that's gone over the course of two periods, each of which has a period of about 12.

00:26 So we're going to write out a periodic curve in the form negative a times cosine of b t minus c plus d in something that will approximate this curve.

00:40 So for now, i'm going to remove this curve so that this graph so that we can start finding what a, b, c, and d are going to look like.

00:50 So let's start out with a.

00:52 So a corresponds with the amplitude.

00:54 So in your textbook, you found that a equals the maximum minus the minimum divided by two.

01:04 So that's going to be the maximum, here is 105 .1, minus the minimum, which is 65 .5 ,5, divided by two.

01:18 So that's going to be about 39 .6 divided by 2, which is equal to 19 .8.

01:28 Is equal to 19 .8.

01:31 Next, b.

01:32 So b corresponds with the period.

01:36 So the period is equal to 12 in this case, because we're going over the course 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12 months.

01:50 So our period is 12.

01:52 That's why we repeated it twice on our plot.