Greenwood press

50
LINEAR FUNCTIONS

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supply so that an electrical object can operate correctly. Resistors are small
devices that block or slow down the current so that an object does not receive too
much power. For example, if the resistance in a light circuit is too low, then the
bulb would receive an overload of power and be destroyed. If the resistance is
too high, then there will not be enough power reaching the bulb in order for it to
light well. These problems can arise with some appliances when they are moved
to different countries, because the electrical circuits may run with different cur-
rent levels. Consequently, appliances may have different types of resistors so that
they can accommodate to the corresponding current levels in a circuit.
In a business setting, a linear function could be used to relate the total costs
needed to sell a product in terms of the number of products produced. For exam-
ple, suppose a bakery created cookies at a raw material expense of $0.25 per
cookie. Suppose production costs for equipment are an extra $500. In this case,
the linear function t = 0.25c + 500 will represent the total cost, t, needed to pro-
duce c cookies. In general, if a function is modeled by a linear relationship, then
the rate ($0.25 per cookie) will be the slope, and the starting amount ($500 equip-
ment expense) will be the y-intercept of the equation. This information is useful
to the owner, because he or she will be able to predict the average cost of pro-
ducing cookies, start-up expenses included, or the amount of cookies that can be
produced based on a fixed budget. 
Unit conversions are often linearly related. For example, the United States
uses a different temperature scale (Fahrenheit) than most of the rest of the world
(Celsius). If an individual from the United States travels to Spain, then a tem-
perature of 30° Celsius would feel considerably different from a temperature of
30° Fahrenheit. The equation that converts the two variables can be determined
by using the freezing and boiling points of water. Water freezes at 0° Celsius and
32° Fahrenheit; water boils at 100° Celsius and 212° Fahrenheit. These two
pieces of information represent two ordered pairs on a line, (0,32) and (100,212).
Since two points are sufficient information to determine the equation of a 
line, the slope formula and y-intercept will lead to the equation 
F =
9
5
C + 32,
where
F is the temperature in Fahrenheit, and C is the temperature in Celsius.
This means that a report of 30° weather in Spain suggests that the day could be
spent at the beach, while in the United States a report of 30° weather means that
you might be having snow! 
Linear functions can be used to form relationships between data that are
found in natural events and places. For example, there is a strong relationship
between the winning time of the men’s Olympic 100-meter dash and the year in
which it occurs. The graph that follows shows that a line can be drawn to approx-
imate the relationship between these two variables. Notice that all of the data val-
ues do not fall on the line, but instead cluster around it. It is possible for points
to be away from the line, especially during years of unusually exceptional per-
formance. The correlation coefficient, r, is a measure of the strength of the lin-
ear relationship. The relationship is stronger as the absolute value of the correla-
tion coefficient approaches the value of 1. If the correlation coefficient is closer
to 0, then a linear relationship does not likely exist. In the 100-meter dash situa-