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Exploring Functions: Characteristics, Applications, and Real-Life Examples

This unit delves into the characteristics and applications of functions, focusing on increasing and decreasing behavior, maxima and minima, asymptotes, end behavior, and continuity. Students will learn the technical definitions and common language associated with these concepts, supported by graphical representations. With practical applications, including tracking heart rate changes following medication, students will utilize graphing calculators to analyze function behavior, identify critical points, and explore piecewise-defined functions. This comprehensive approach bridges theory with real-world scenarios.

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Exploring Functions: Characteristics, Applications, and Real-Life Examples

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  1. Unit 1 Characteristics and Applications of Functions

  2. Parent Function Checklist Unit 1: Characteristics and Applications of Functions

  3. Parent Function Checklist

  4. Parent Function Checklist

  5. Parent Function Checklist

  6. Parent Function Checklist

  7. Parent Function Checklist

  8. Function Vocabulary Unit 1: Characteristics and Applications of Functions

  9. Increasing • Picture/Example • Common Language: Goes up from left to right. • Technical Language: f(x) is increasing on an interval when, for any a and b in the interval, if a > b, then f(a) > f(b).

  10. Decreasing • Picture/Example • Common Language: Goes down from left to right. • Technical Language: f(x) is decreasing on an interval when, for any a and b in the interval, if a > b, then f(a) < f(b).

  11. Maximum • Picture/Example • Common Language: Relative “high point” • Technical Language: A function f(x) reaches a maximum value at x = a if f(x) is increasing when x < a and decreasing when x > a. The maximum value of the function is f(a).

  12. Minimum • Picture/Example • Common Language: Relative “low point” • Technical Language: A function f(x) reaches a minimum value at x = a if f(x) is decreasing when x < a and increasing when x > a. The minimum value of the function is f(a).

  13. Asymptote • Picture/Example • Common Language: A boundary line • Technical Language: A line that a function approaches for extreme values of either x or y.

  14. Odd Function • Picture/Example • Common Language: A function that is symmetric with respect to the origin. • Technical Language: A function is odd iff f(-x) = -f(x).

  15. Even Function • Picture/Example • Common Language: A function that has symmetry with respect to the y-axis • Technical Language: A function is even iff f(-x)=f(x)

  16. End Behavior • Picture/Example • Common Language: Whether the graph (f(x)) goes up, goes down, or flattens out on the extreme left and right. • Technical Language: As x-values approach ∞ or -∞, the function values can approach a number (f(x)n) or can increase or decrease without bound (f(x)±∞).

  17. Heart Medicine Unit 1: Characteristics and Applications of Functions

  18. In the function editor of your calculator enter:

  19. Table

  20. Graph

  21. 1) Use a graphing calculator to find the maximum rate at which the patient’s heart was beating. After how many minutes did this occur? • 79.267 beats per minute • 1.87 minutes after the medicine was given

  22. 2) Describe how the patient’s heart rate behaved after reaching this maximum. • The heart rate starts decreasing, but levels off. • The heart rate never drops below a certain level (asymptote).

  23. 3) According to this model, what would be the patient’s heart rate 3 hours after the medicine was given? After 4 hours? • 3 hours = 180 minutes  h(180) ≈ 60.4 bpm • 4 hours = 240 minutes  h(240) ≈ 60.3 bpm

  24. 4) This function has a horizontal asymptote. Where does it occur? How can it’s presence be confirmed using a graphing calculator? • Asymptote: h(x)=60 • Scroll down the table and look at large values of x or trace the graph and look at large values of x. • The end behavior of the function is: As x  ∞, f(x)  60 and as x  -∞, f(x)  60

  25. End Behavior Unit 1: Characteristics and Applications of Functions

  26. End Behavior

  27. End Behavior

  28. End Behavior

  29. End Behavior

  30. End Behavior

  31. End Behavior

  32. Piecewise-Defined Functions Unit 1: Characteristics and Applications of Functions

  33. Evaluate the function at the given values by first determining which formula to use.

  34. Define a piecewise function based on the description provided.

  35. Graph the given piecewise functions on the grids provided.

  36. Continuity Unit 1: Characteristics and Applications of Functions

  37. Continuity-Uninterrupted in time or space.

  38. 1) Complete the table and answer the questions that follow.

  39. 2) Graph each function using a “decimal” window (zoom 4) to observe the different ways in which functions can lack continuity.

  40. 3) Graph each function to determine where each discontinuity occurs. Classify each type.

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