Table of Contents
Understanding Linear Functions
A linear function is one of the simplest and most important types of functions in algebra. It describes a straight-line relationship between two variables, usually called $x$ (input) and $y$ (output).
A function $f$ is called linear (in this school-level sense) if its graph is a straight line and it can be written in the form
$$
y = mx + b
$$
or
$$
f(x) = mx + b.
$$
Here:
- $m$ is the slope (how steep the line is and in which direction it goes).
- $b$ is the $y$-intercept (where the line crosses the $y$-axis).
The detailed meanings of “slope” and “intercept form” belong to their own subsections; here we focus on recognizing, using, and interpreting linear functions as a whole.
Recognizing Linear Functions
A function is linear (in this context) if:
- Its equation can be written as $y = mx + b$, where $m$ and $b$ are constants.
- Each variable appears only to the first power (no $x^2$, $\sqrt{x}$, $\dfrac{1}{x}$, etc.).
- The graph is a straight line.
Examples of linear functions:
- $y = 3x - 5$
- $f(x) = -\dfrac{1}{2}x + 4$
- $y = 7$ (this is linear with $m = 0$, $b = 7$)
Not linear:
- $y = x^2 + 3$ (because of $x^2$)
- $y = \dfrac{1}{x}$ (because of $x$ in the denominator)
- $y = \sqrt{x} + 1$ (because of the square root)
A quick test: if you can rearrange the equation to look like $y = mx + b$ without introducing powers, roots, or products of variables, then it is linear.
Different Forms of Linear Equations
Even though $y = mx + b$ is the most familiar form, linear functions can be written in several algebraically equivalent ways. All of them still represent straight lines.
Slope–intercept form (function form)
This is the standard “functional” way to write a linear function:
$$
y = mx + b \quad\text{or}\quad f(x) = mx + b.
$$
It shows directly:
- $m$: how $y$ changes when $x$ increases by 1.
- $b$: the value of $y$ when $x = 0$.
For example, $f(x) = 2x + 3$:
- $m = 2$ (for every increase of 1 in $x$, $y$ increases by 2),
- $b = 3$ ($f(0) = 3$).
Point–slope idea
Sometimes a line is given by:
- a point $(x_1, y_1)$ on the line, and
- a slope $m$.
Then the equation can be written in point–slope form:
$$
y - y_1 = m(x - x_1).
$$
This is especially useful for building a linear function from information like “the line passes through $(2, 5)$ with slope $-3$.”
You can always rearrange point–slope form into $y = mx + b$.
Standard form
A linear equation in two variables can also be written as
$$
Ax + By = C,
$$
where $A$, $B$, and $C$ are constants and $A$ and $B$ are not both zero.
Examples:
- $2x + 3y = 6$
- $-x + 4y = 10$
Any such equation (with $B \neq 0$) can be solved for $y$ to give you a slope–intercept form, so it still represents a linear function.
Linear Functions as Rules of Change
One way to understand a linear function is to think of it as a rule describing:
- a starting value, and
- a constant rate of change.
In $y = mx + b$:
- $b$ is the starting value (when $x = 0$),
- $m$ is the constant rate of change (how $y$ changes for each unit change in $x$).
Example: $y = 50x + 20$ might describe the cost $y$ (in dollars) of renting a bike for $x$ hours, where:
- the basic fee is \$20 (starting value),
- the cost increases by \$50 for each additional hour (constant rate of change).
Linear Functions from Tables
A function is linear if its table of values shows a constant change in $y$ for equal changes in $x$.
For example:
| $x$ | $y$ |
|---|---|
| 0 | 3 |
| 1 | 7 |
| 2 | 11 |
| 3 | 15 |
Here, each time $x$ increases by 1, $y$ increases by 4. The rate of change is constant, so this is linear.
To find the equation:
- Constant change in $y$ is $4$, so $m = 4$.
- When $x = 0$, $y = 3$, so $b = 3$.
Thus, $y = 4x + 3$.
If the change in $y$ is not constant, the function is not linear.
Linear Functions in Context (Models)
Linear functions are often used to model real-world relationships that change at a constant rate. Typical examples include:
- Constant speed: distance vs. time when speed is steady.
- Example: $d(t) = 60t$, where $d$ is distance in kilometers, $t$ is time in hours. The distance increases by 60 km every hour.
- Earning money: wages vs. hours worked for a fixed hourly pay.
- Example: $E(h) = 15h$, where $E$ is earnings in dollars, $h$ is hours worked, at \$15 per hour.
- Temperature conversion: between Celsius and Fahrenheit.
- Example: $F(C) = \dfrac{9}{5}C + 32$.
When you build a linear model from words, you typically:
- Identify the starting value (value at $x = 0$).
- Identify the constant rate of change.
- Write $y = mx + b$ using these two numbers.
Interpreting Coefficients in Linear Models
Given a linear function in context, interpret $m$ and $b$ in words:
- $b$ (the intercept) often answers:
“What is the value when the input is zero?” - $m$ (the slope) often answers:
“How much does the output change when the input increases by 1?”
Example: $P(t) = 2000 + 150t$ might represent population $P$ of a town $t$ years after 2020.
- $b = 2000$: The population at $t = 0$ (year 2020) is 2000 people.
- $m = 150$: The population increases by 150 people per year.
Comparing Linear Functions
When you have two or more linear functions, you can compare them in terms of:
- Rate of change (slope $m$): which one grows or shrinks faster?
- Starting value (intercept $b$): which one starts higher or lower?
For example, suppose:
- Plan A: $C_A(x) = 10x + 5$
- Plan B: $C_B(x) = 8x + 15$
Interpretation:
- Plan A has a higher rate of change ($10$ vs. $8$), so it gets more expensive more quickly as $x$ increases.
- Plan B has a higher starting cost ($15$ vs. $5$) when $x = 0$.
In many problems, you might be asked which plan is cheaper for small $x$ or for large $x$; this comparison comes from understanding the slopes and intercepts of the linear functions.
When Linear Functions Are Appropriate
Linear models are appropriate when:
- The relationship appears roughly like a straight line when graphed.
- Data suggests a constant rate of change.
- The situation described uses phrases like “per hour,” “per item,” “each year,” with no indication that this rate itself changes.
They are not appropriate when the rate of change itself is changing noticeably, such as compound interest (handled by exponential functions), or motion with acceleration.
Understanding linear functions as “straight-line rules with constant rate of change” prepares you to:
- Work with their slope,
- Use intercepts,
- Graph them quickly,
- And translate between equations, tables, graphs, and word descriptions in the later subsections devoted to those specific skills.