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Characteristics and Applications of Functions Unit 1: Characteristics and Applications of Functions Parent Function Checklist Parent Function Checklist Parent Function Checklist Parent Function Checklist Parent Function Checklist Unit 1: Characteristics and Applications of Functions 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). 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). 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). 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). Asymptote Picture/Example Common Language: A boundary line Technical Language: A line that a function approaches for extreme values of either x or y. 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). 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) 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)±∞). Unit 1: Characteristics and Applications of Functions In the function editor of your calculator enter: y (12(5x 6x 1)) /( x 1) 2 2 Table Graph 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 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). 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 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 Unit 1: Characteristics and Applications of Functions End Behavior End Behavior End Behavior End Behavior End Behavior End Behavior Unit 1: Characteristics and Applications of Functions Evaluate the function at the given values by first determining which formula to use. Define a piecewise function based on the description provided. Graph the given piecewise functions on the grids provided. Unit 1: Characteristics and Applications of Functions Continuity- Uninterrupted in time or space. 1) Complete the table and answer the questions that follow. 2) Graph each function using a “decimal” window (zoom 4) to observe the different ways in which functions can lack continuity. 3) Graph each function to determine where each discontinuity occurs. Classify each type.