Lesson 3: Linear Functions and Rates of Change

Property

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

where $b$ is the initial or starting value of the function (when input, $x = 0$), and $m$ is the constant rate of change, or slope of the function. The $y$-intercept is at $(0, b)$.

Examples

• A car travels at a constant speed of 50 miles per hour. Its distance $D$ from a starting point after $t$ hours can be modeled by $D(t) = 50t$.
• A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C(x) = 0.05x + 20$.
• For the function $f(x) = 3x + 2$, the value when $x=4$ is $f(4) = 3(4) + 2 = 14$. The point $(4, 14)$ is on the line.

Explanation

Think of a linear function as a rule for anything that changes at a steady rate. The 'm' is the rate of change (how steep the line is), and 'b' is the starting point on the vertical axis before any change happens.

Property

To find the rate of change ($m$) and initial value ($b$) from a representation:

• From a Graph: The rate of change is the slope, $m = \frac{{\text{rise}}}{{\text{run}}} = \frac{{\Delta y}}{{\Delta x}}$. The initial value, $b$, is the y-coordinate of the point where the line crosses the y-axis, $(0, b)$.
• From a Table: Select any two points $(x_1, y_1)$ and $(x_2, y_2)$ to find the rate of change, $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$. The initial value, $b$, is the value of $y$ when $x=0$.

Examples

Property

In a real-world linear model written in the form $y = mx + b$, the mathematical variables have specific, practical meanings:

• The slope ($m$) represents the rate of change. It describes how much the dependent variable ($y$) changes for every one-unit increase in the independent variable ($x$).
• The y-intercept ($b$) represents the initial value or starting point. It is the value of the dependent variable ($y$) when the independent variable ($x$) is 0.

Examples

• An equation for manatee deaths is $y = 4.7 + 2.6t$, where $t$ is years since 1975. This line models an increasing trend, with deaths increasing by about 2.6 per year.
• A plumber charges a fee based on $C = 75h + 50$, where $C$ is total cost and $h$ is hours worked. The slope is 75, meaning the cost increases by 75 dollars for each hour of work. The y-intercept is 50, meaning there is a 50 dollar initial fee before any work begins.
• The amount of water $V$ in a tank after $t$ minutes is modeled by $V = -10t + 300$. The slope is -10, meaning the water decreases by 10 gallons each minute. The y-intercept is 300, meaning the tank initially contained 300 gallons.

Explanation

When a real-world situation is modeled by a linear function, the slope and y-intercept are no longer just abstract numbers. The slope tells you the exact rate at which a quantity is changing over time or per unit. The y-intercept tells you the starting amount or fixed fee before that change even begins.