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.