MLC at Boise State 2013
A Cubic Regression
Group Activity 4
Business Project Week #7
In the first activity we looked at a set of data that was modeled by a line (a linear regression). In
the second and third activities we looked at data that was modeled by a quadratic (a quadratic
regression). In the last activity we also looked more in depth at characteristics of a quadratic. In
this activity we are going to model a third degree polynomial that is also called a cubic.
Looking at profits:
A business sells hot chocolate and ice cream at a local
football stadium. Last year the business kept track of profits
on a weekly basis and the amounts are listed to the right.
Using Temperature for the domain and Profit for the range
values, describe a good set of values for window settings:
x-min =
x-max =
x-scl =
y-min =
y-max =
y-scl =
Using these scale settings and the data in the chart, sketch a
graph of the data.
Profits for last year’s games:
Week
Temperature
Profit
1
2
3
4
5
6
7
8
9
10
11
12
102
83
97
66
75
30
22
60
54
-10
-1
8
$915
$256
$1087
-$403
-$123
$375
$926
-$403
-$393
$3406
$4067
$2454
MLC at Boise State 2013
Notice the data has a shape. What polynomial equation do your think will best fit the data?
Using your calculator, determine the equation for a quadratic model of the data. The quadratic
model is?
Build a second model, a cubic model. The cubic model is?
Looking at your sketch, what type of regression best fits your data? (linear, quadratic, cubic,
exponential) Explain.
Graph the data points, your quadratic model and your cubic model on the calculator at the same
time. Determine if you think the quadratic or the cubic model is better and explain why.
Using the model and the data, determine an implied domain. The implied domain is the domain
where you would use the model to make future estimations. Notice that you can put 1000 degrees
into your equation and obtain an answer but hopefully we realize that at 1000 degrees nobody
would be at the game and the answer would be absurd. So what is your implied domain? This
does not have an exact answer and each group may have different endpoints but they should be
close.
Using your model, determine the interval of temperatures where the business will lose money.
Answer:
Using your model, determine the temperature, which will create the maximum profit.
Answer:
Using your model, estimate the profit to be made if the temperature is going to be 28 degrees?
Answer:
MLC at Boise State 2013
New problem: Repeat the above steps for the following data.
Profits for last year’s games:
Week
Temperature
Profit
1
2
3
4
5
6
7
8
9
10
95
77
68
66
39
30
22
55
18
-10
$1715
$556
$187
$103
$123
$475
$826
-$43
$1133
$3806
Connections:
Relating previously learned skills to a new skill: We plan to provide at least one problem on each
test which expands previously learned skills to a new application. Today we will be factoring
higher degree polynomials and exploring the relationship between the polynomial factors, the
zeros on the graph and function values.
This week you are going to use the remainder theorem. When you use the remainder theorem you
are finding the remainder when a polynomial is divided by a binomial. If the remainder is zero,
there are a few things that we know about our polynomial, its factored form and its graph.
Let’s start with the polynomial f x   x 4  8x 3  17 x 2  2 x  24 .
Find f 2 .
MLC at Boise State 2013
On your calculator and on the grid below, graph f x   x 4  8x 3  17 x 2  2 x  24 .
Make a connection between the graph of f x  and the value found for f 2 .
Looking at the graph of f x  , what other values for x will give a remainder of zero?
Now, we will factor f x  using the conclusions reached above. What factor corresponds to the xintercept of 2?
Use the graph above for f x  , factor f x  .
Now, use the graph of f x  , to determine when f x   0 . Indicate this solution on your graph.
MLC at Boise State 2013
Now let’s look at the equation 8x 3 + 24 = x 4 +17x 2 + 2x ; list two ways that you can solve this
equation graphically. Then choose the best method and solve it graphically.
How would you solve the equation algebraically? Solve the equation algebraically and check your
solution with your graphical solution.
More examples:
Solve the following equations both graphically and algebraically.
1. Graph and solve 13x  10  x 3  2 x 2 . Then factor the expression x 3  2 x 2  13x  10 .
2. Graph and solve 10 x 3  50 x  x 4  35x 2  24 . Then factor the expression
x 4 10 x 3  35x 2  50 x  24 .