Chapter 1 Section 1
Modeling and Equation
Solving
What You’ll Learn About & Why
Numerical Models
 Algebraic Models
 Graphical Models
 The Zero Factor Property
 Problem Solving
 Grapher Failure and Hidden Behavior
 Why?

◦ Numerical, Algebraic, & Graphical Models
provide different methods to visualize, analyze,
and understand data.
Mathematical Models

Scientists and Engineers use mathematics to
model the real world and thereby unravel the
mysteries of the universe.

A mathematical model is a structure that
approximates phenomena for the purpose of
studying and predicting their behavior.

We will be concerned with 3 types of
mathematical models: numerical, graphical, &
algebraic.
Mathematic Models

If you would like to predict the expected
rainfall total over the next year, which type
of model would be most helpful to you?
◦ Numerical (rainfall totals over the past 10 yrs)
◦ Graphical (a scatter plot of rainfall totals vs. yrs)
◦ Algebraic (a formula for yearly rainfall totals)
Numerical Models

This is the most basic of the 3 models, in which
numbers (or data) are placed into a table and
analyzed to gain insights into phenomena.

Numerical models can be as simple as baseball
statistics or as complicated as the network of
interrelated numbers that measure the global
economy.

See Table 1.1 on page 70
Graphical Models

From the numerical data, we can create a graph to
represent the data.

This usually comes as a scatter-plot relating the
number of objects (y) to a time period (x)

This helps us to visualize the change in behavior
and identify what is happening over time.

See Figures 1.1, 1.2, & 1.3 on page 73.
Algebraic Models
We can use the graphical model and technology to
create an algebraic model, i.e. find a line of best fit
using a regression analysis.
 Or we can use formulas that are known to us to
solve a problem by using algebra.
 For example, we can compare the areas of
rectangular pizza to circular pizza by using
formulas
 See Example 3 on page 71.

Example

Let’s look at some problems on
www.interactmath.com to see how we can
connect the numerical, graphical, and
algebraic models.
Zero Factor Property

Definition: A product of real numbers is zero iff at
least one of the factors in the product is zero.

If a b c = 0, then one of the following is true:
a = 0, b = 0, or c = 0.

We use this concept to solve a number of
equations that are factorable.
Zero Factor Property


Example: Find all real numbers x for which
6x3 = 11x2 + 10x
Solution: Begin by setting the equation equal
to 0, and solve by factoring.
3
2
6 x  11x  10 x  0


x 6 x  11x  10  0
2
x2 x  53x  2  0
 The zero factor property then says that either
x = 0, or 2x – 5 = 0, or 3x + 2 = 0.
 Which yields that x = 0, or x = 5/2, or x = -2/3
as our solutions.
Fundamental Connection

If a is a real number that solves the equation
f(x) = 0, then the following 3 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). Which is the coordinate point (a,0).
Problem Solving by George Pólya
George Pólya (1887 – 1985) is considered
to be the “father” of modern problem
solving because of his book How to Solve
It: A New Aspect of Mathematical Method.
 His Four-Step Process is incredibly simple
yet very effective.

1.
2.
3.
4.
Understand the Problem
Devise a Plan
Carry out the plan
Look Back
Problem Solving

See the table on pages 76 – 77 for a clearer
view of Pólya’s method.
Example of Applying the Process

The engineers at an auto manufacturer pay
students $0.08 per mile plus $25 per day to
road test their new vehicles.
1. How much did the auto manufacturer pay
Sally to drive 440 miles in one day?
2. John earned $93 test-driving a new car in one
day. How far did he drive?
Example of Applying the Process
How would you model these 2 problems?
 How would you solve the problems?
 How could you support the answers?
 How do you interpret the answers?

A note about Graphing Calculators
While graphing calculators are wonderful
tools to help understand and “visualize” the
algebraic models we are studying,
sometimes they “lie” and will give you false
information.
 For example, sometimes in a quadratic
function, a graphing calculator will “show”
an x-intercept when there really is not one.
Why? Because the graph may be so close
that from a “distance” it looks like it
touches.

Example

Look at y = 3/(2x – 5)
◦ Does the function cross the x-axis?
Example

Look at y = 3/(2x – 5)
◦ Does the function cross the x-axis?
 No, the graphing calculator “draws” vertical
asymptotes (imaginary lines that the function
never crosses) and gives the illusion that the
graph exists at the point x = 5/2.
 Algebraically, y must equal 0 for an x-intercept
to exist. Notice:
Example
Algebraically, y must equal 0 for an x-intercept
to exist. Notice:
3
y
2x  5
3
0
2x  5
0(2 x  5)  3
03
false
Another Example

View the following function on the
Geometer’s Sketchpad or your graphing
calculator and see if you can solve it.
 x3
– 1.1x2 – 65.4x + 229.5 = 0
Homework
Start with # 3 – 30 by multiples of 3.
 Tomorrow do # 33 – 60 by multiples of 3.
