Proportional Models:  or  and :  or 
 

A Direct Proportional relationship is one which an increase in the input of the independent variable or attribute results in an increase in the output or dependent variable or attribute by a constant ratio called k, the constant of proportionality. or  k and n are constants, n > 0

As x increases y also increases so we have an increasing function. Often students need to observe the value of the function as x increases to find clues as to the nature and parameters of a function.

The parameters are the constants values of a function that determine its shape, slope, transformation, limits, and sketching boundaries. Some parameters are natural occurring values such as .

Table 1.5 Example 3 Directly Proportional Problem:
 

Word Problem Statement: The Circumference, C of a circle is directly proportional to the radius of the circle, from the table of values find the formula for the Circumference of a circle.	Table of Values:     Radius Circumference 0 0 2 12.5664 4 25.2327
Graph from Table: Figure 1.5	Formula from table or graph: Since directly proportional:  Or so  So formula is: C = 6.2832 r

Note the key word or phrase (directly proportional) in the word problem that identify the problems as a 
direct proportional mathematical model. You should look of key word or phrase when trying find the clues
that describe the type of mathematical model being considered.
 

A Inverse Proportional relationship is one which an increase in the input of the independent variable or attribute results in a decrease in the output or dependent variable or attribute by a constant ratio called k, the constant of proportionality. or  k and n are constants, n > 0

As x increases y decreases, so we have a decreasing function. This type of function is also called a 
Rational Function. Rational functions are characterized by functions that contain asymptotes and holes. 
 

Asymptotes are lines where the function tends to approach as values for the independent variable becomes very large or very small. Holes are regions or points where the function does not exist (that is, no x or y values)

Table 1.6 Example 4 Inverse Proportional Problem:
 

Word Problem Statement: The Population in thousands is inversely proportional to the square of the time in years since 1985 and is given by values in the table on right. Find a formula for the Population size, P with time, t.	Table of Values:     Time, t, years since 1985 Population in 1000 2 1.25 3 0.5556 4 0.3125
Graph from Table: Figure 1.6	Formula from table or graph: Since directly proportional:  Or so  So formula is:

One set of points is needed to find the formula for proportional problems.

Percent:
 

Percent is a value that indicates parts per hundred:  So 45% =  0.45 is the decimal equivalent of 45%

In this text it is often required to convert percent to its decimal equivalence and visa versa especially when we
work with exponential models. See learning module on Basic Math for more about percents.

Rates and Rate of Change:

In a world were the only constant thing is change, the rate of change becomes an important measure of how
fast or slow things are changing. Rates are proportions of two quantities or the ratio of two attributes or variable that are changing.
 

Rate is the ratio of two variables or quantities, A and B that are changing and is often stated as the ratio of the change in quantity A over the change in quantity B. Sometimes the rate is referred to as the rate of change and is model by: Rate

Examples of rates includes:

Speed: rate of change of distance with time:  (55 mph)

Population Growth: rate of change of population size with time:  (10,000 per year)

Pay: rate of pay, $ with hours worked:  ($12 per hour)

Currency Exchange: rate of  per $: 

Percent: parts per hundred: