Greenwood press

LINEAR FUNCTIONS
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means that the ratio of these changes, called the slope, is also constant. For exam-
ple, the previous comparison is the same as saying that there will be a change of
fifteen units in the dependent variable for every change in three units of the inde-
pendent variable, since this ratio simplifies to 5. Every linear function can be
written in the slope-intercept form, 
y = mx + b, where m is the slope of the line,
and
b is in the y-intercept.
Realistic situations use linear functions to make predictions or draw compar-
isons that involve constant change. For example, the cost of gasoline is linearly
related to the number of gallons purchased. For every one gallon of gas pur-
chased, the price will increase approximately $1.40. The fact that the gas price
per gallon does not change as gas is pumped allows someone to use a linear func-
tion to predict the amount of money needed to fill the tank. In this situation, the
function c = 1.40g would relate the cost in c dollars to g gallons purchased. If an
automobile has a twelve-gallon tank, then the cost to fill the tank would be c =
1.40(12) = $16.80. In addition, the linear equation is useful when the individual
purchasing gasoline would like to know how much gasoline he or she would
obtain with the $10 available in his or her pocket. In this case, 10 would be sub-
stituted for the variable c, and solving the equation would show that approxi-
mately 7.14 gallons could be purchased, slightly more than half a tank in most
cars.
Linear functions are useful in estimating the amount of time it will take to
complete a road trip. Assuming that traffic conditions are good and the driver is
traveling at a constant speed on a highway, the linear equation d = rt (distance
equals rate times time) can be used to predict the total distance traveled or time
needed to complete the trip. For example, suppose that a family is traveling on
vacation by automobile. The family members study a map to determine the dis-
tance between the cities, estimate a highway speed or rate of 65 miles per hour,
and then solve the linear equation d = 65t to estimate the length of their trip. An
awareness of the time needed for the trip would likely help the family plan a time
of departure and times for rest stops.
Banking institutions determine the amount of simple interest accumulated on
an account by using the linear equation I = Prt, where I is the amount of interest,
P is the initial principal, r is the interest rate, and t is the time in years in which
the interest has been accumulating. For example, a $1,000 loan with 8 percent
simple interest uses the function I = 1000(0.08)t, or simplified to I = 80t, to pre-
dict the amount of interest over a specific time period. Once the principal and
interest rates have been determined, the function is linear, since the amount of
interest increases at a constant rate over time. Over five years, there will be I =
80(5) = $400 net payment in interest. 
Circuits rely on linear relationships in order to operate electrical equipment.
The voltage V, current I, and resistance R are related with the equation V = IR. A
power supply has voltage to create a stream of current through electrical wires.
The current in a circuit is typically held constant, such as at 72 Hz, so that there
is a constant stream of electricity. In this case, the linear relationship V = 72R
would help a manufacturer determine the amount of resistance needed in a power