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The table shows the average daily temperature on the first day of each month for one year.

Use a graphing calculator to graph the data points. Use 0 for January, 1 for February, 2 for March, and so on.

The graph of these points follows a path resembling a
[tex]\[
\begin{tabular}{|l|c|}
\hline \multicolumn{1}{|c|}{ Month } & Temperature $\left({ }^{\circ} F \right)$ \\
\hline January & 23 \\
\hline February & 35 \\
\hline March & 48 \\
\hline April & 65 \\
\hline May & 74 \\
\hline June & 80 \\
\hline July & 76 \\
\hline August & 64 \\
\hline September & 50 \\
\hline October & 32 \\
\hline November & 24 \\
\hline December & 20 \\
\hline
\end{tabular}
\][/tex]

A. cosine curve reflected across the [tex]$y$[/tex]-axis
B. sine curve reflected across the [tex]$y$[/tex]-axis
C. cosine curve reflected across the [tex]$x$[/tex]-axis
D. sine curve reflected across the [tex]$x$[/tex]-axis



Answer :

GinnyAnswer
To determine the pattern in the temperature data, we need to graph the temperatures against the months.

Let's break down the given data:

[tex]\[ \begin{array}{|c|c|} \hline \text{Month} & \text{Temperature} ({}^\circ F) \\ \hline 0 & 23 \\ \hline 1 & 35 \\ \hline 2 & 48 \\ \hline 3 & 65 \\ \hline 4 & 74 \\ \hline 5 & 80 \\ \hline 6 & 76 \\ \hline 7 & 64 \\ \hline 8 & 50 \\ \hline 9 & 32 \\ \hline 10 & 24 \\ \hline 11 & 20 \\ \hline \end{array} \][/tex]

When these points are graphed, you will notice that the temperatures tend to increase from January (0) to June (5), peak in June, and then start to decrease from July (6) to December (11). This pattern is characteristic of a trigonometric curve.

To determine the specific type of trigonometric pattern, observe the following:

1. The temperature initially increases, reaches a maximum, and then decreases, resembling the general shape of a sine or cosine wave.
2. Comparing the pattern:

- A sine wave [tex]\( \sin(x) \)[/tex] starts at 0, rises to its maximum, falls through 0, and then continues to its minimum before returning to 0.
- A cosine wave [tex]\( \cos(x) \)[/tex] starts at its maximum, decreases to its minimum, and then returns to its maximum.

3. Given that our data starts at a point not at the maximum or minimum but rather lower and rises to the peak, it is not precisely aligned with the regular sine or cosine starting points.

4. Adjusting the cosine wave by reflecting [tex]\( \cos(x) \)[/tex] across the x-axis (transforming it into [tex]\(-\cos(x)\)[/tex]) resembles our temperature pattern more closely—starting at a lower point (January's temperature) and peaking mid-year before declining again.

Therefore, the observed pattern with the points resembles a "cosine curve reflected across the [tex]\( x \)[/tex]-axis."

So, the correct answer is:

[tex]\[ \boxed{\text{cosine curve reflected across the } x\text{-axis}} \][/tex]

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