What is a linear function?

It is a function given by f(x) = ax + b, where a and b are numbers. Its graph is always a straight line. The number a is called the slope and the number b the intercept.

What does the slope a mean?

It is the rate of change: by how much the value changes when the argument grows by one. With a = 3 every step to the right lifts the graph by three units. On the graph a decides how steep the line is, and its sign decides whether the function increases or decreases.

What does the intercept b mean?

It is the value of the function at zero, f(0) = b. On the graph it is the height at which the line crosses the y-axis — the starting value from which the change described by a begins.

How do I find the zero of a linear function?

By solving ax + b = 0, which gives x₀ = −b/a. The formula works whenever a ≠ 0. For f(x) = 2x − 6 we get x₀ = 6/2 = 3, so the line crosses the x-axis at (3, 0).

When is a linear function increasing and when decreasing?

Increasing when a > 0, decreasing when a < 0. When a = 0 it is neither — it is constant, and its graph is a horizontal line. Only the sign of a decides; the intercept b has nothing to do with monotonicity.

How do I find the line through two points?

First compute the slope a = (y₂ − y₁)/(x₂ − x₁), then substitute the coordinates of one point into y = ax + b and solve for b. For A(1, 3) and B(3, 7) we get a = 4/2 = 2 and 3 = 2 · 1 + b, so b = 1 and f(x) = 2x + 1.

How do I turn the general form into the slope–intercept form?

Just solve for y. From 2x − y + 3 = 0 we get −y = −2x − 3, and multiplying both sides by −1 gives y = 2x + 3. The change is possible whenever the coefficient of y is non-zero — otherwise the equation describes a vertical line, which is not the graph of any function.

When are two lines parallel and when perpendicular?

Parallel when their slopes are equal (a₁ = a₂), perpendicular when the product of the slopes is −1 (a₁ · a₂ = −1). So a line perpendicular to y = 2x + 5 has slope −1/2.

What is direct proportionality?

It is the special case of a linear function with b = 0, that is y = ax. The graph then passes through the origin and the ratio y/x is constant. Distance at a constant speed behaves like this, and so does the cost at a constant price per item.

Intermediate level

Linear function

Q: How do I find the line through two points?

First compute the slope a = (y₂ − y₁)/(x₂ − x₁), then substitute the coordinates of one point into y = ax + b and solve for b. For A(1, 3) and B(3, 7) we get a = 4/2 = 2 and 3 = 2 · 1 + b, so b = 1 and f(x) = 2x + 1.

Q: How do I turn the general form into the slope–intercept form?

Just solve for y. From 2x − y + 3 = 0 we get −y = −2x − 3, and multiplying both sides by −1 gives y = 2x + 3. The change is possible whenever the coefficient of y is non-zero — otherwise the equation describes a vertical line, which is not the graph of any function.

Q: When are two lines parallel and when perpendicular?

Parallel when their slopes are equal (a₁ = a₂), perpendicular when the product of the slopes is −1 (a₁ · a₂ = −1). So a line perpendicular to y = 2x + 5 has slope −1/2.

Q: What is direct proportionality?

It is the special case of a linear function with b = 0, that is y = ax. The graph then passes through the origin and the ratio y/x is constant. Distance at a constant speed behaves like this, and so does the cost at a constant price per item.

A linear function y = ax + b draws a straight line. Meet the meaning of the slope and the intercept, the zero, the formula of a line through two points, the general form, and the conditions for parallel and perpendicular lines.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

What a function isA function is an unambiguous assignment: every argument gets exactly one value. Meet the f(x) notation, the domain and the range, the four ways to define a function, and the vertical line test on a graph.
Linear equationsA linear equation is an equality with the unknown in the first power. Learn the operations that keep equations equivalent, solve one step by step, check the result, rearrange a formula to make another quantity its subject, and handle an unknown in a denominator.

Where this is used

Real situations where you count exactly the way this lesson teaches:

The electricity bill
A bill comes in two parts: standing charges that do not depend on usage, and a price per kilowatt-hour. With 38 of standing charges and 0.92 per kWh the bill is f(x) = 0.92x + 38, so 210 kWh costs 231.20. The intercept explains the thing that surprises people most: twice the usage is not twice the bill — 420 kWh comes to 424.40, which is 1.84 times as much, not two.
Convert it here: Energy
Calibrating a pressure sensor
An industrial transmitter returns a current of 4 to 20 mA across a range of 0 to 16 bar. The engineer builds the formula from two points: a = (16 − 0)/(20 − 4) = 1 bar per milliamp, and the intercept is set so that 4 mA reads zero, giving p(I) = I − 4. A reading of 12.5 mA is 8.5 bar. Without that line the controller sees a current and knows nothing about what is happening in the pipe.
Convert it here: Pressure
Choosing a gym plan
The gym offers 149 a month unlimited, or 39 plus 18 per visit. The second offer is the line 18x + 39, the first is the horizontal line 149. They meet where 18x + 39 = 149, that is x = 6.1, so the flat plan starts paying from the seventh visit of the month. The slope says how fast the cost climbs; the crossing point says when the decision flips.

All formulas

Slope–intercept form
$f(x) = ax + b$
a — the slope, b — the intercept
Slope from two points
$a = \frac{y_2 - y_1}{x_2 - x_1}$
the rise divided by the run
The zero
$x_0 = -\frac{b}{a} \quad (a \neq 0)$
where the graph crosses the x-axis
The value at zero
$f(0) = b$
where the graph crosses the y-axis
General form
$Ax + By + C = 0$
reducible to the slope–intercept form whenever B ≠ 0
Condition for parallel lines
$a_1 = a_2$
parallel lines have equal slopes
Condition for perpendicular lines
$a_1 \cdot a_2 = -1$
hence a₂ = −1/a₁

A linear function is a function given by

f(x) = ax + b

where $a$ and $b$ are fixed numbers. Its graph is — always, without exception — a straight line. The number $a$ is the slope, the number $b$ the intercept.

What a and b mean

Both numbers have a plain meaning, and the graph shows it at once:

$b$ is the starting value: $f(0) = b$ , the height at which the line crosses the $y$ -axis;
$a$ is the rate of change: by that much the value grows when the argument grows by one.

The line f(x) = 2x − 3: it crosses the y-axis at −3, and every step of one to the right lifts it by two.

The dashed guide on the drawing is exactly how the slope is read off: one right, two up. Hence the other, equally good definition of $a$ — the rise divided by the run:

a = \frac{y_2 - y_1}{x_2 - x_1}

Monotonicity

Whether the function increases is decided by the sign of $a$ alone:

Three linear functions: increasing (a > 0), decreasing (a < 0) and constant (a = 0).

$a > 0$ — the function increases, the line rises to the right;
$a < 0$ — the function decreases, the line falls;
$a = 0$ — the function is constant, $f(x) = b$ , a horizontal line.

The intercept $b$ has nothing to do with monotonicity — it slides the line up or down without changing how steep it is.

The zero

The zero is the argument whose value is zero, that is the solution of $ax + b = 0$ :

x_0 = -\frac{b}{a} \qquad (a \neq 0)

Find the zero of f(x) = 2x − 6 and both axis crossings of its graph.

When $a = 0$ there is no point in using the formula: a constant function $f(x) = b$ either has no zeros at all (for $b \neq 0$ ) or takes the value zero everywhere (for $b = 0$ ).

The line through two points

Two points determine exactly one line, so from them the formula can be recovered. First the slope, then the intercept:

Find the linear function whose graph passes through A(1, 3) and B(3, 7).

The order of subtraction in the numerator and the denominator must be the same — otherwise the sign comes out reversed. If the numerator starts from $y_2$ , the denominator starts from $x_2$ .

The general form

The same set of points can be described by an equation in which $y$ does not stand alone on the left:

Ax + By + C = 0

That is the general form. To read the slope off it, just solve for $y$ :

Turn 4x − 2y + 6 = 0 into the slope–intercept form.

The change is only possible when $B \neq 0$ . The equation $x = 3$ describes a vertical line, and such a line — as we know from the vertical line test — is not the graph of any function.

Parallel and perpendicular lines

Only the coefficient $a$ carries the steepness of a line, so both conditions on the mutual position of two lines are about it:

a_1 = a_2 \quad \text{(parallel)}, \qquad a_1 \cdot a_2 = -1 \quad \text{(perpendicular)}

Perpendicular lines: the product of their slopes is 2 · (−0.5) = −1.

The condition gives a handy formula: a line perpendicular to one of slope $a$ has slope

a_\perp = -\frac{1}{a}

the reciprocal with the opposite sign. For $a = 3$ that is $-\frac{1}{3}$ ; for $a = -\frac{2}{5}$ it is $\frac{5}{2}$ .

Direct proportionality

When $b = 0$ the formula collapses to $y = ax$ and the graph passes through the origin. The ratio $\frac{y}{x}$ is then constant and equal to $a$ — this is direct proportionality.

Distance at a constant speed behaves like that ( $s = vt$ ), and so does the cost of buying at a fixed price per item. When $b \neq 0$ a "joining fee" appears: a subscription added to a bill, the flag-fall of a taxi, the standing charge on an electricity invoice.

A taxi charges a fixed fee of 8 plus 3 per kilometre. Write the cost as a function and find after how many kilometres it passes 50.

Exercises

One session covers everything this lesson teaches: the zero, the slope from two points, the whole formula from two points, the general form turned into the slope–intercept one, and the slope of a perpendicular line. Type a formula in any correct spelling — 2x+1, 1+2x and 2*x+1 all mean the same thing.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

f(x) = 2x − 6 → x₀

Common mistakes

Swapping $a$ and $b$ — the steepness is $a$ , the crossing with the $y$ -axis is $b$ .
The sign of the zero — the formula gives $x_0 = -\frac{b}{a}$ , not $\frac{b}{a}$ .
Subtracting in opposite orders — numerator and denominator must start from the same point.
Dividing only one term — when solving for $y$ , the whole side of the equation is divided.
Perpendicularity as a bare reciprocal — the sign has to change too: $a_\perp = -\frac{1}{a}$ .
Calling a vertical line a linear function — $x = 3$ is not the graph of a function.

Formula card

Topic: Linear function

Slope–intercept form
$f(x) = ax + b$
a — the slope, b — the intercept
Slope from two points
$a = \frac{y_2 - y_1}{x_2 - x_1}$
the rise divided by the run
The zero
$x_0 = -\frac{b}{a} \quad (a \neq 0)$
where the graph crosses the x-axis
The value at zero
$f(0) = b$
where the graph crosses the y-axis
General form
$Ax + By + C = 0$
reducible to the slope–intercept form whenever B ≠ 0
Condition for parallel lines
$a_1 = a_2$
parallel lines have equal slopes
Condition for perpendicular lines
$a_1 \cdot a_2 = -1$
hence a₂ = −1/a₁

The slope is the rise per unit of argument: one step right, two steps up — a = 2.

The sign of the slope decides the monotonicity: a > 0 increasing, a < 0 decreasing, a = 0 constant.

Frequently asked questions

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min