Lesson 2: Linear Functions and Slope

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 $D = 50t$.
• A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C = 0.05x + 20$.
• For the function $y = 3x + 2$, 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

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope $m$ of the line is calculated using the formula:

Examples

Property

In a real-world context described by the equation $y = mx + b$:

• The slope ($m$) represents the rate of change. It tells you 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 zero.

Examples

Property

Important things to remember about linear functions are:

• If the line intersects the y-axis in the point $(0, b)$, then the equation of the line is $y = mx + b$.
• If the line is horizontal, the slope is zero, and the equation of the line is $y = b$.
• If the line is vertical, it has no slope, and its equation is $x = a$.
• If the line goes through the origin, the equation of the line is $y = mx$ and the values of $y$ are proportional to the values of $x$; otherwise said, $y/x = m$.
• If the line has slope $m$, and the point $(x_0, y_0)$ is on the line, then the equation of the line is $y - y_0 = m(x - x_0)$.
• If $(x_0, y_0)$, $(x_1, y_1)$ are on the line, then a point $(x, y)$ is on the line if $\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}$.

Examples

• A line has a slope of 4 and a y-intercept at $(0, -5)$. Its equation is $y = 4x - 5$.
• A horizontal line passes through the point $(3, 8)$. Since the slope is 0, its equation is $y = 8$.
• A line passes through $(2, 1)$ with a slope of 3. Using the point-slope form, the equation is $y - 1 = 3(x - 2)$, which simplifies to $y = 3x - 5$.

Explanation

A linear function has a constant rate of change, its slope. This means its graph is a straight line. Unlike proportional relationships, a linear function doesn't have to go through the origin, which is what the 'b' in $y=mx+b$ represents.