Unit 1 Notes
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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.