Linear Functions are Important. They are everywhere!
One reason that linear functions are so important is that in the
simplest contexts where two variable measurements are made, the
relationship between those variables are best described by a linear
Where do you find Linear Functions? The contexts for finding linear
functions are all around us.
On the road:
When we drive our cars (or bicycles), walk, or run at a constant
speed on a road , the distance we travel is a linear function of
time. If we let $s$ denote the distance along the road we have
traveled, $t$ denote the time that has elapsed, and $r$ denote our
rate of travel, then $s = r \times t$, "distance is rate times
The amount of gas consumed by a car is usually estimated as a linear
function of the distance traveled.
If we let $g$ denote the amount of gas consumed on a trip, $s$
denote the distance along the road we have traveled, $r_g$ denote
the rate at which gas is consumed by our car, then $g = r_g \times s
$, "gas is rate of consumption times distance."
At the store:
When we purchase a commodity (vegetables, rice, oil, gasoline) at
the market by weight or volume, the cost is a linear function of the
amount being purchased.
If we let $C$ denote the cost of the commodity being purchased, $x$
denote the amount of the commodity we are buying, and $p$ denote the
unit price of the commodity, then $C = p \times x.$, "cost is price
times amount purchased."
When we sell a commodity (vegetables, rice, oil, gasoline) at the
market by weight or volume, the revenue is a linear function of the
amount being sold.
If we let $R$ denote the revenue received for the commodity being
sold, $x$ denote the amount of the commodity sold, and $p$ denote
the unit price of the commodity, then $R = p \times x$, "revenue is
price times amount sold."
When we convert measurements from the English to the metric system ,
the metric measurement is a linear function of the the English
measurement. For example, if we let $F$ denote the temperature
measured in the Fahrenheit scale and $C$ denote the same temperature
measured in the Centigrade scale, then $F = \frac 9 5 C + 32$, a
When a object is moving freely in a fall above the surface of the
earth, its velocity is a linear function of time.
If we let $v$ denote the velocity of the moving object, $t$ denote
the time the object has been in free fall and $g$ denote the
acceleration of the object due to the gravitational force of the
earth, then $v = g \times t$, "velocity is acceleration times