Lesson 9-4: Describe Proportional and Nonproportional 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

A graph represents a non-proportional relationship if it fails at least one of the two visual requirements for proportionality:

1. The graph is not a straight line.
2. The graph does not pass through the origin $(0,0)$.

Examples

Property

In real-world application problems, the $x$- and $y$-intercepts represent the value of one quantity when the other quantity is zero.

Examples

An equation for a fundraiser is $5x + 10y = 500$, where $x$ is student tickets and $y$ is adult tickets.

• The $x$-intercept is 100, meaning 100 student tickets are sold if zero adult tickets are sold.
• The $y$-intercept is 50, meaning 50 adult tickets are sold if zero student tickets are sold.

An equation for a school sports event is $3x + 4y = 120$, where $x$ is the number of boys participating and $y$ is the number of girls participating.

• The $x$-intercept is 40, meaning 40 boys participate if zero girls participate.
• The $y$-intercept is 30, meaning 30 girls participate if zero boys participate.

Explanation

Intercepts tell a story of extremes! The $x$-intercept reveals what happens when you have “none” of the item on the $y$-axis, while the $y$-intercept shows what occurs when you have “none” of the item on the $x$-axis. It's a great way to understand the limits of a situation.