Lesson 2: Representing Linear Relationships

Property

A linear relationship models a constant rate of change between two quantities and can be represented by an equation of the form $y = mx + b$.
In this relationship, the change in the output variable ($y$) is proportional to the change in the input variable ($x$).
The graph of a linear relationship is a straight line that intercepts the y-axis at the starting value, $b$.

Examples

• A taxi charges a 3 dollars flat fee plus 2 dollars per mile. The cost $C$ for a trip of $d$ miles is given by the linear equation $C = 2d + 3$. The graph starts at $(0,3)$.
• A new phone plan costs 30 dollars a month, which includes 5 gigabytes of data, plus 10 dollars for each additional gigabyte. The cost $C$ for $g$ gigabytes over 5 is $C = 30 + 10(g-5)$.
• The temperature in Fahrenheit ($F$) is a linear function of the temperature in Celsius ($C$), given by $F = \frac{9}{5}C + 32$. The y-intercept is 32, which is the freezing point in Fahrenheit.

Explanation

Think of a linear relationship as a proportional one with a head start. You begin at a starting value ($b$), and then add a constant amount ($m$) for every step. The graph is a straight line, but it starts at $b$, not zero.

Property

In a linear relationship, the rate of change between the two variables is constant.
For any two points, the ratio of the change in the vertical variable to the change in the horizontal variable is always the same.
This constant rate of change is the slope of the line, represented by $m$ in the equation $y = mx + b$.
A positive slope indicates an increasing relationship, while a negative slope indicates a decreasing one.

Examples

• In the equation $h = 100 - 8t$ for the height of a draining tank, the rate of change is -8. This means the water level decreases by 8 inches every minute.
• A plant's height is modeled by $H = 1.5w + 5$, where $w$ is weeks. The rate of change is 1.5, meaning the plant grows 1.5 centimeters taller each week.
• A salesman's monthly salary is $S = 1500 + 200c$, where $c$ is the number of cars sold. The rate of change is 200, meaning his salary increases by 200 dollars for each car he sells.

Explanation

The rate of change, or slope, tells you the steepness of the line. It's how much the $y$ value changes for every one-unit increase in the $x$ value. A negative rate means it's a downhill slope; $y$ decreases as $x$ increases.