Math Problem Statement

The graph represents the monthly average temperature in degrees Fahrenheit from January to December, modeled by a polynomial function f. The turning points are approximately (3,70) and (9,20). Interpret these turning points.

Solution

Let's analyze the problem step by step based on the information from the image:

Part (a): Estimate the turning points.

  • Turning points are where the polynomial function changes direction, meaning it shifts from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum).

    Based on the graph:

    • The first turning point occurs around x=3x = 3, where the temperature reaches a high of about 70F70^\circ F. So the first turning point is approximately (3,70)(3, 70).
    • The second turning point occurs around x=9x = 9, where the temperature reaches a low of about 20F20^\circ F. So the second turning point is approximately (9,20)(9, 20).

    Therefore, the turning points are (3,70)(3, 70) and (9,20)(9, 20).

Part (b): Interpret the turning points.

  1. Interpret the turning point with the low monthly average temperature:

    • The second turning point, (9,20)(9, 20), represents the low monthly average temperature.
    • Since x=9x = 9 corresponds to September, and the temperature is about 20F20^\circ F, the correct interpretation is C: "The low monthly average temperature of 20F20^\circ F occurs in September."
  2. Interpret the turning point with the high monthly average temperature:

    • The first turning point, (3,70)(3, 70), represents the high monthly average temperature.
    • Since x=3x = 3 corresponds to March, and the temperature is about 70F70^\circ F, the correct interpretation is A: "The high monthly average temperature of 70F70^\circ F occurs in March."

To summarize:

  • Turning points: (3,70)(3, 70) and (9,20)(9, 20)
  • Low temperature: 20F20^\circ F in September (option C)
  • High temperature: 70F70^\circ F in March (option A)

Would you like more details on the graph interpretation or polynomial functions?


Here are five related questions to extend your understanding:

  1. How do you find the turning points of a polynomial function analytically?
  2. What is the significance of the first and second derivatives in identifying turning points?
  3. How can the degree of a polynomial function influence the number of turning points?
  4. How would you estimate the area under the curve of this temperature graph?
  5. What are some real-life applications of turning points in fields like economics or physics?

Tip: When interpreting graphs, turning points often indicate significant transitions, such as peaks in temperature or other phenomena in nature!

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Math Problem Analysis

Mathematical Concepts

Graphing polynomial functions
Turning points
Local maxima and minima
Function interpretation

Formulas

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Theorems

Basic polynomial behavior

Suitable Grade Level

Grades 9-12