Linear functions – understand y = kx + m

3.1The concept of a function

A function is a rule that takes a value and returns one particular value. Think of a machine: you put in a number x, and the machine returns the number f(x).

For example, if f(x) = 2x + 3, then:

f(0) = 2·0 + 3 = 3
f(1) = 2·1 + 3 = 5
f(5) = 2·5 + 3 = 13

For each x there is one and only one value f(x).

The vertical line test

Draw a vertical line through the graph:

If it crosses the graph at at most one point for any x value → it is a function.
If it crosses at two or more points → it is not a function.

A circle, for example, is not a function, because a vertical line crosses it at two points.

Domain and range

Two important concepts when discussing functions:

The domain (D_f): all the x values the function can take.
The range (V_f): all the y values the function returns.

Example: For f(x) = √x, we need x ≥ 0, so D_f = [0, →⟩ and V_f = [0, →⟩. For g(x) = 1/x, x cannot be 0, so D_g = all numbers except 0.

The graph of a function

The graph consists of all the points (x, f(x)) plotted on coordinate axes.

To read the graph:

Choose an x value on the x-axis.
Go straight up (or down) to the graph.
Go horizontally to the y-axis — that is f(x).

Example: at x = 1, f(1) = 5, so the point (1, 5) lies on the graph.

Table of values

A table of values shows a selection of (x, f(x)) pairs. Example for f(x) = x² − 2:

x	−2	−1	0	1	2
f(x)	2	−1	−2	−1	2

Tip: A table of values is useful when drawing a graph by hand. Choose 5–7 sensible x values around 0.

3.3Linear functions

A linear function has a straight-line graph. General form:

f(x) = a·x + b

a = slope (how steep the line is)
b = constant term (intersection with the y-axis)

The slope between two points (x₁, y₁) and (x₂, y₂):

a = (y₂ − y₁) / (x₂ − x₁)

a > 0: the line rises
a < 0: the line falls
a = 0: horizontal line

Point–slope form

If you know one point (x₁, y₁) and the slope a:

y − y₁ = a · (x − x₁)

Example: find the line through (1, 2) and (3, 8)

Find the slope a. a = (8 − 2) / (3 − 1) = 6/2 = 3

Use point–slope form. With (1, 2) as the known point. y − 2 = 3·(x − 1)

Simplify. y = 3x − 3 + 2 = 3x − 1

Answer: the line is y = 3x − 1.

Linear growth models

Use a linear function when something increases or decreases by the same amount per unit of time (constant change). Examples: a car travelling at a constant speed, a pool filling at a constant rate, a monthly fee.

3.2Intersections

Intersections are points where the graph crosses an axis or where two graphs cross each other.

Intersection with the y-axis

Set x = 0 in the function expression. This gives the y coordinate where the graph crosses the y-axis.

Intersection with the y-axis = (0, f(0))

Example: f(x) = 3x − 6 ⇒ f(0) = −6. The graph crosses the y-axis at (0, −6).

Intersections with the x-axis (roots)

Set f(x) = 0 and solve for x. These x values are called roots.

Intersections with the x-axis: solve f(x) = 0

Example: f(x) = x² − 4 ⇒ x² − 4 = 0 ⇒ x = ±2. The roots are at (−2, 0) and (2, 0).

Not every graph has roots! For f(x) = x² + 1, we get x² = −1, which has no solution among the real numbers. The graph lies entirely above the x-axis and never crosses it. That is perfectly valid — it simply means that f has no roots.

Intersections between two graphs

Two graphs meet where f(x) = g(x). Solve the equation for x, then find the y value by substituting x into either function.

Intersection of f and g: solve f(x) = g(x)

Example: f(x) = 2x + 1 and g(x) = −x + 4. Set them equal: 2x + 1 = −x + 4 ⇒ 3x = 3 ⇒ x = 1. Then y = f(1) = 3. They intersect at (1, 3).

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