Linear functions are widely used in modeling real world applications. Whether it is trying to determine the most cost effective mobile phone plan or trying to predict the weather in the coming days, linear functions have many applications in our everyday lives. Linear functions, or lines, describe relationships where the rate of change is constant. For example, the speed of a car is increasing at a constant rate of $\rm 1.2~km$ per second or the price of a stock on the New York Stock Exchange is decreasing at a rate of $€0.15$ per minute. Linear functions describe how something is changing per unit of time. Whether it is liters per minute or kilometers per second, the important thing to remember is, associated with every linear function is a rate of change and that rate of change is always the same, or constant.

Example 1: Determine the equation of the line that goes through the points $(−4,1)$ and $(3,15)$.

Explanation: We must first determine the slope which can be calculated as.

$m=\dfrac{15−1}{3−(−4)}=\dfrac{14}{7}=2$. Using the point-slope form of a line and choosing either point gives:

$y−1=2(x−(−4))$
$\Rightarrow y−1=2(x+4)$
$\Rightarrow y=2x+9$

which is the equation of the line in slope-intercept form.

Example 2: The cost of a rental car can be given by $\mathrm C(k)=0.08k+15$ where $\mathrm C(k)$ is the cost, in Euros to rent the car after driving $k$ kilometers. Determine the significance of the numbers $0.08$ and $15$ in the function.

Explanation: $0.08$ can be interpreted as the slope of the line, or rate of change. Since we are talking about Euros and kilometers, it will cost $€0.08$ to drive $1$ kilometer. The value of $15$ is the cost to rent the car after driving $\rm 0~km$. In other words, it is the flat rate cost to rent the car.