Differentiate between linear and exponential functions.
 Linear functions take the form
y = mx + b.
 An exponential function is in
the form y = ax.
 It is easy to see the difference
between a linear and
exponential function on a
graph.
 Linear functions
change at a constant
rate per unit interval.
 An exponential
function changes by
a common ratio over
equal intervals.
x
y=2x
x
y=2x
1
2
1
2
2
4
2
4
3
6
3
8
4
8
4
16
5
10
5
32
6
12
6
64
7
14
7
128
8
16
8
256
9
18
9
512
10
20
10
1,024
 Sebastian deposits $100,000 in a local bank that will pay out 5% interest every year.
Linear or exponential? Why? Show evidence.
 A certain type of corn grows at the rate of 3 inches per week. Linear or exponential?
Why? Show evidence.
 The Munn Sugar Processing Plant is able to process 10 tons of sugar per month.
Linear or exponential? Why? Show evidence.
 Exercise biologist, Samantha, discovered that to reduce soreness, people should
start biceps curls at 10 pounds. Then, progress weekly to 11 pounds, 14 pounds, 20
pounds, 32 pounds, 56 pounds and so on. Linear or exponential? Why…
 The growth of bacteria doubles every hour. Linear or exponential? Why…
 A person sends out an email to 8 individuals, who each then passes it on to 8 other
individuals, and each of these individuals passes it on to 8 other individuals, and so
on. Linear or exponential? Why…
Continuous
Discrete
Definition: A set of data is said to be continuous if the
values belonging to the set can take on ANY value within a
finite or infinite interval.
Definition: A set of data is said to be discrete if the values
belonging to the set are distinct and separate (unconnected
values).
Examples:
• The height of a horse (could be any value within the range of
Examples:
• The number of people in your class (no fractional parts of a
horse heights).
person).
• Time to complete a task (which could be measured to fractions of
• The number of TV sets in a home (no fractional parts of a TV
seconds).
set).
• The outdoor temperature at noon (any value within possible
temperatures ranges.)
• The number of puppies in a liter (no fractional puppies).
• The number of questions on a math test (no incomplete
• The speed of a car on Route 3 (assuming legal speed limits).
questions).
NOTE: Continuous data usually requires a measuring
device. (Ruler, stop watch, thermometer, speedometer,
etc.)
NOTE: Discrete data is counted. The description of the
task is usually preceded by the words "number of...".
Function: In the graph of a continuous function, the points
are connected with a continuous line, since every point has
meaning to the original problem.
Function: In the graph of a discrete function,
only separate, distinct points are plotted, and only these
points have meaning to the original problem.
Graph: You can draw a continuous function without
lifting your pencil from your paper.
Graph: A discrete graph is a series of unconnected points
(a scatter plot).
Domain: a set of input values consisting ofall numbers in
an interval.
Domain: a set of input values consisting of only certain
numbers in an interval.
In Plain English: A continuous function allows the xvalues to be ANY points in the interval, including
fractions, decimals, and irrational values.
In Plain English: A discrete function allows the x-values
to be only certain points in the interval, usually only
integers or whole numbers.
Why do we care? When graphing a function, especially one related to a real-world situation, it is important to choose
an appropriate domain (x-values) for the graph. For example, if a function represents the number of people left on an
island at the end of each week in the Survivor Game, an appropriate domain would be positive integers. Hopefully, half
of a person is not an appropriate answer for any of the weeks. The graph of the people remaining on the island would be
a discrete graph, not a continuous graph.
IS THE EXAMPLE A DISCRETE OR CONTINUOUS
MODEL? Why…
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The average person takes 12,000 steps a day.
The shoe sizes of Stadium students.
The average rain fall in November in the city of Tacoma.
The hair of a poodle grows ¼ inch per month.
The number of marbles in a jar.
The city of Seattle adds 5,000 tons of trash to its landfills every day.
For every ton of paper that is recycled, 17 trees are saved.
The range of temperature for next week.
The number of slices of pizza in a whole pizza.
Stock market prices fell 2% last week.