A meteorologist makes a graph of the low temperatures of the previous week and the low temperatures of the current week to see if a pattern exists. He marks the days of the previous week negative part of the x-axis and the current days on the positive part of the x-axis. He marks the temperatures (in Fahrenheit) on the y-axis. Next, he creates a polynomial function based on t and uses the function to estimate the temperature pattern under similar conditions.

Answer 1

It sounds like the meteorologist is using a visual representation of temperature data to identify patterns and create a polynomial function that can model the temperature changes over the given periods. Here’s a brief overview of how he might proceed:

Step 1: Data Collection

The meteorologist would start by collecting data for the low temperatures from the previous week (let's say days -7 to -1 for the previous week) and the current week (days +1 to +7 for the current week). For example:

• Previous Week (Negative X-axis):

  • Day -7: 30°F
  • Day -6: 32°F
  • Day -5: 28°F
  • Day -4: 35°F
  • Day -3: 31°F
  • Day -2: 29°F
  • Day -1: 33°F

• Current Week (Positive X-axis):

  • Day +1: 34°F
  • Day +2: 36°F
  • Day +3: 30°F
  • Day +4: 29°F
  • Day +5: 31°F
  • Day +6: 32°F
  • Day +7: 33°F

Step 2: Plotting the Data

Using the temperature data, he could create a scatter plot where the x-axis represents the days (with negative values for last week and positive values for this week) and the y-axis represents temperature in Fahrenheit:

• Plot (-7, 30), (-6, 32), (-5, 28), ..., (7, 33).

Step 3: Creating the Polynomial Function

The meteorologist could then use polynomial regression or curve fitting methods to find a polynomial that best fits the plotted points. This polynomial function \( P(t) \) could take the general form:

\[ P(t) = a_n t^n + a_{n-1} t^{n-1} + ... + a_1 t + a_0 \]

where \( t \) represents time (the days relative to the current week) and \( a_n, a_{n-1}, ..., a_0 \) are constants determined through regression analysis.

Step 4: Analyzing the Function

Once the polynomial function is created, the meteorologist can use it to estimate temperatures for future days under similar conditions. If, for example, the polynomial is found to be:

\[ P(t) = -0.2t^3 + 1.5t^2 + 32 \]

He could input values for \( t \) to predict future low temperatures based on previous trends.

Step 5: Identifying Patterns

Finally, by analyzing the characteristics of the polynomial (such as its degree, contributions of positive/negative coefficients, and behavior at the extremes), the meteorologist can look for patterns, such as:

• Whether temperatures generally increase or decrease through the weeks.
• Fluctuations in temperature, indicating weather variability.
• Anomalies (e.g., peaks or troughs) that might warrant further investigation.

Conclusion

By using this approach, the meteorologist can better understand temperature patterns and potentially make more accurate forecasts in the context of observed historical data. This analysis can also be beneficial in understanding how certain weather conditions might correlate with temperature trends.