1. Mathematical Models
Includes…geometric formulas, regression analysis, solving equations in 1 variable

2. Definitions A mathematical model is a mathematical
structure that approximates phenomena for the
purpose of studying or predicting their behavior
One type of mathematical model: numerical
model, where numbers (or data) are analyzed to
gain insights into phenomena

3. period during which the
Table 1.1 The Minimum Hourly Wage
Year Min. Hourly Purchasing Power
Wage in 2001 Dollars
In what year did a
minimum-wage worker
have the greatest
purchasing power?
In 1970
What was the longest
period during which the
minimum wage did not
increase?
From ,
, and

4. nearly twice as much as a
Table 1.1 The Minimum Hourly Wage
Year Min. Hourly Purchasing Power
Wage in 2001 Dollars
A worker making min.
wage in 1980 was earning
nearly twice as much as a
worker making min. wage
in 1970  so why was there
pressure to once again
raise the min. wage?
Purchasing power
actually dropped by
$0.43 during that
period (inflation)

5. Definitions Another type of mathematical model:
Algebraic Model – uses formulas to relate
variable quantities associated with the
phenomena being studied
(Benefit: can generate numerical values of unknown
quantities using known quantities)

6. Guided Practice
At Dominos, a small (10” diameter) cheese pizza costs $4.00,
while a large (14” diameter) cheese pizza costs $8.99.
Assuming that both pizzas are the same thickness, which is
the better value?
Calculate areas per dollar cost:
Small Pizza
Large Pizza 2 2
Small: in /$, Large: in /$
The small pizza is the better value!!!

7. Regression Analysis: Graphical Model – visual representation of a
numerical or algebraic model that gives insight
into the relationships between variable quantities
Regression Analysis:
The process of analyzing data by creating a scatter
plot, critiquing the data’s appearance (linear,
parabolic, cubic, etc.), choosing the appropriate
model, finding the line of best fit, making predictions
about the data.

8. A Good Example:
Galileo gathered data on a ball rolling down an inclined plane:
Elapsed Time (seconds) 1 2 3 4 5 6 7 8
Distance Traveled (in)
.75
6.75 12
18.75 27
36.75 48
1. Create a scatter plot of these data
2. Derive an algebraic model to fit these data 2
d = 0.75t
3. Graph this function on top of your scatter plot

9. A Good Example: d = 168.75 in t = 9.092 sec
Galileo gathered data on a ball rolling down an inclined plane:
Elapsed Time (seconds) 1 2 3 4 5 6 7 8
Distance Traveled (in)
.75
6.75 12
18.75 27
36.75 48
4. How far will the ball have traveled after 15 seconds?
d = in
5. How long will it take the ball to travel 62 inches?
t = sec

10. Terminology:
If a is a real number that solves the equation f(x) = 0, then
these three statements are equivalent:
1. The number a is a root (or solution) of the equation
f(x) = 0.
2. The number a is a zero of y = f(x).
3. The number a is an x-intercept of the graph of y = f(x).
(sometimes the point (a, 0) is referred to as an x-intercept)

11. More Practice Problems…3 2
Find all real numbers x for which 6x = 11x x 5 2
x = 0 or x = or x = – 2 3
We just used the Zero Product Property:
A product of real numbers is zero if and only if at least one
of the factors in the product is zero.

12. Guided Practice
Solve the equation algebraically and graphically.

13. Guided Practice Solve the equation algebraically and graphically.
Use the quadratic formula:
Check for extraneous solutions!!!

14. Practice 3 Find all x-intercepts of y = 2x – 5
Solving algebraically???
(there are no x-intercepts)
Solving graphically???

15. Try another problem: x = –9, 5, 5.1
Solve graphically: x – 1.1x – 65.4x = 0 3 2
x = –9, 5, 5.1
This is an example of hidden behavior – when
an inaccurate viewing window obscures details
of a graph

16. Now, on to some basic problem solving…
The engineers at an auto manufacturer pay students $0.08 per
mile plus $25 per day to road test their new vehicles.
1. Derive an algebraic model for the students’ pay.
p = 0.08x + 25
(where p = students’ pay,
and x = miles driven)

17. Now, on to some basic problem solving…
The engineers at an auto manufacturer pay students $0.08 per
mile plus $25 per day to road test their new vehicles.
2. How much did the auto manufacturer pay Sally to drive
440 miles in one day?
$60.20
3. John earned $93 test-driving a new car in one day. How far
did he drive?
850 miles

18. Now, on to some basic problem solving…
A math student’s grade is determined by weights, with home-
work counting 20% and quizzes/tests counting 80%.
Derive an algebraic model for the student’s grade, given that
homework and quizzes/tests are graded by points.
Total HW pts.
Total Q/T pts.
g =
Poss. HW pts.
Poss. Q/T pts.

19. Now, on to some basic problem solving…
A math student’s grade is determined by weights, with home-
work counting 20% and quizzes/tests counting 80%.
2. What is Wolfgang’s grade if he has earned 26 out of a
possible 28 homework points, quiz grades of 22/25 and 13/15,
and a test grade of 41/50?
Grade =
86.127%

20. Now, on to some basic problem solving…
A math student’s grade is determined by weights, with home-
work counting 20% and quizzes/tests counting 80%.
3. If Jan has perfect homework scores and quiz scores of 20/25
and 14/15, what does she need to earn on the upcoming
50-point test in order to have an overall 90% grade average?
At least
44.75 points

21. Now, on to some basic problem solving…
Solve the given equation graphically by converting it to an
equivalent equation with 0 on the right-hand side and then
finding the x-intercepts. 1. 2.
Homework: Odds p.76 #7-11, 15, 17, 21, 29-47