Unit 1

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# Unit 1 - PowerPoint PPT Presentation

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

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### Unit 1

Characteristics and Applications of Functions

Parent Function Checklist

Unit 1: Characteristics and Applications of Functions

Function Vocabulary

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)±∞).
Heart Medicine

Unit 1: Characteristics and Applications of Functions

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
End Behavior

Unit 1: Characteristics and Applications of Functions

Piecewise-Defined Functions

Unit 1: Characteristics and Applications of Functions

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

Unit 1: Characteristics and Applications of Functions

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