A2M3L22 SB ­ Choosing a Model.notebook
Choosing a Model
February 26, 2016
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
You are working on a team analyzing the following
data points gathered by your colleagues:
(­1.5, 5)
(0,105)
(1.5,178)
(4.3, 120)
How can we
determine what type
of equation best
models this data?
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
The points fit the quadratic
model shown at the right .....
but... this sinusoidal
model fits the points
also....
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
Suppose the four points
represent positions of a
projectile fired into the air.
Which of the two model is
more appropriate in that
situation? Why?
Choosing the correct model depends on the context and what we know about what the data represents.
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
Some things to think about when deciding on a model:
­ the end behavior of the function
­ how the function changes ­ as input increases by 1 unit how does the output change
­ relative minimums and maximums
­ range of the function
Characteristics of non­constant linear functions:
A2M3L22 SB ­ Choosing a Model.notebook
Characteristics of quadratic functions:
Characteristics of exponential functions:
February 26, 2016
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
Characteristics of sinusoidal functions:
An artist is design posters for a new advertising campaign. The first
poster takes 10 hours to design, but each subsequent poster takes
roughly 15 minutes less time than the previous one as he gets more
practice.
What type of function models the amount of time needed to
create n posters for n < 20?
Write find a model that is appropriate for this situation.
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
A homeowner notices that her heating bill is the lowest in the month
of August and increases until it reaches its highest amount in the
month of February. After February, the amount of the heating bill
slowly drops back to the level it was in August again. The amount
of the bill in February is roughly four times the amount of the bill in
August.
What type of function models the amount of the heating
bill in a particular month? How do you know?
What other information is needed to create a model of
this scenario?
The amount of caffeine in a subject's bloodstream decreases by half
every 3.5 hours. A latte contains 150 mg. of caffeine, which is
absorbed into the bloodstream almost immediately.
What type of function models the caffeine level in the
patient's bloodstream at time t hours after drinking the
latte?
Find a model that is appropriate for this situation.
A2M3L22 SB ­ Choosing a Model.notebook
February 26, 2016
Lesson Summary
§ If we expect from the context that each new term in the sequence of data
is a constant added to the previous term, then we try a linear model.
§ If we expect from the context that the second differences of the sequence
are constant (meaning that the rate of change between terms either grows or
shrinks linearly), then we try a quadratic model.
§ If we expect from the context that each new term in the sequence of data
is a constant multiple of the previous term, then we try an exponential model.
§ If we expect from the context that the sequence of terms is periodic, then
we try a sinusoidal model.
Model
Equation of Function
Rate of Change
Linear
Constant
Quadratic
Changing linearly
Exponential
mult. of currrent value
Sinusoidal
Periodic
Homework: Worksheet
Quiz tomorrow
A2M3L22 SB ­ Choosing a Model.notebook
y=2x
x
y
­3
­6
­2
­4
­1
­2
0
0
1
2
2
4
3
6
February 26, 2016
y = x2
>
>
>
>
>
>
­2
­2
­2
­2
­2
­2
x
y
­3
9
­2
4
­1
1
0
0
1
2
3
1
4
9
>
>
>
>
>
>
5
3
1
­1
­3
­5
>
>
>
>
>
2
2
2
2
2