Linear Function

Article Contents

What is a Linear Function?
Slope k
Y-intercept q
Drawing the Graph (Two-Point Method)
SVG: Comparing Linear Functions
Special Cases
Parallel Lines
Finding the Equation from Two Points
Intersection with Axes
Properties
Common Mistakes
Interactive Exercises

1. What is a Linear Function? {#what-is-linear}

A linear function is a function of the form:

f (x) = k x + q

where:

$k$ is the slope (gradient) -- how steep the line is
$q$ is the y-intercept -- where the line crosses the y-axis

The graph is always a straight line.

Relationship to Direct Proportion

When $q = 0$ , the linear function becomes $f (x) = k x$ , which is a direct proportion. So direct proportion is a special case of the linear function.

2. Slope k {#slope-k}

The slope $k$ tells us how much $y$ changes when $x$ increases by 1:

k = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Value of $k$	Meaning	Graph
$k > 0$	Function is increasing	Line goes up to the right
$k < 0$	Function is decreasing	Line goes down to the right
$k = 0$	Function is constant	Horizontal line ( $y = q$ )
$	k	$ large	Steep line
$	k	$ small	Gentle line

Example

For $f (x) = 3 x + 1$ : the slope is $k = 3$ .

This means: when $x$ increases by 1, $y$ increases by 3.

3. Y-intercept q {#y-intercept-q}

The y-intercept $q$ is the value of $f (0)$ :

f (0) = k \cdot 0 + q = q

So the graph crosses the y-axis at the point $(0, q)$ .

Value of $q$	Where the line crosses the y-axis
$q > 0$	Above the origin
$q = 0$	At the origin (direct proportion)
$q < 0$	Below the origin

4. Drawing the Graph (Two-Point Method) {#drawing-graph}

To draw a straight line, you only need two points. Here is the method:

Step 1: Find two points

Choose two easy values of $x$ (often $x = 0$ and $x = 1$ , or where $f (x) = 0$ ).

Step 2: Calculate $f (x)$

Step 3: Plot and connect

Example: $f (x) = 2 x + 1$

Point 1: $f (0) = 2 (0) + 1 = 1$ gives $(0, 1)$
Point 2: $f (1) = 2 (1) + 1 = 3$ gives $(1, 3)$
Point 3 (check): $f (- 1) = 2 (- 1) + 1 = - 1$ gives $(- 1, - 1)$

Plot these points and draw a straight line through them.

5. SVG: Comparing Linear Functions {#comparing-functions}

Function	Slope $k$	Y-intercept $q$	Behaviour
$y = 2 x + 1$	$2$	$1$	Increasing, steep
$y = - x + 2$	$- 1$	$2$	Decreasing
$y = 0.5 x - 1$	$0.5$	$- 1$	Increasing, gentle

Notice how:

The blue and green lines both go upward (positive $k$ ), but blue is steeper
The red line goes downward (negative $k$ )
Each line crosses the y-axis at its $q$ value

6. Special Cases {#special-cases}

Case	Equation	Graph
$k = 0$	$f (x) = q$	Horizontal line
$q = 0$	$f (x) = k x$	Direct proportion (through origin)
$k = 1, q = 0$	$f (x) = x$	The identity line (45 degree angle)

7. Parallel Lines {#parallel-lines}

Two linear functions are parallel if and only if they have the same slope $k$ .

Example

f (x) = 3 x + 1

and

g (x) = 3 x - 4

are parallel (both have

k = 3

They never intersect -- the distance between them is constant.

Rule: Same slope = parallel. Different slope = the lines will intersect at exactly one point.

8. Finding the Equation from Two Points {#equation-from-points}

Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ :

Step 1: Calculate the slope:

k = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Step 2: Use one point to find

q

q = y_{1} - k \cdot x_{1}

Example

Find the equation of the line through $A (1, 5)$ and $B (3, 11)$ .

k = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3

q = 5 - 3 \cdot 1 = 2

f (x) = 3 x + 2

Check:

f (3) = 3 (3) + 2 = 11

-- correct.

9. Intersection with Axes {#intersection-axes}

Y-axis intersection

Set $x = 0$ :

f (0) = k \cdot 0 + q = q

The line crosses the y-axis at $(0, q)$ .

X-axis intersection (zero of the function)

Set $f (x) = 0$ :

k x + q = 0 ⟹ x = - \frac{q}{k} (k \neq = 0)

The line crosses the x-axis at $(- \frac{q}{k}, 0)$ .

Example: $f (x) = 2 x - 6$

Y-intercept: $(0, - 6)$
X-intercept: $2 x - 6 = 0 ⟹ x = 3$ , so $(3, 0)$

10. Properties {#properties}

Property	Value
Formula	$f (x) = k x + q$
Graph	Straight line
Domain	$D (f) = R$
Range	$R (f) = R$ (if $k \neq = 0$ ) or $R (f) = {q}$ (if $k = 0$ )
Slope	$k = \frac{Δ y}{Δ x}$
Y-intercept	$(0, q)$
X-intercept	$(- \frac{q}{k}, 0)$ for $k \neq = 0$
Increasing	When $k > 0$
Decreasing	When $k < 0$
Parallel lines	Same slope $k$

11. Common Mistakes {#common-mistakes}

Mistake	Correction
Confusing slope and y-intercept	In $y = k x + q$ , $k$ is the slope (multiplies $x$ ), $q$ is the y-intercept (constant)
Using only one point to draw	Use at least two points; three for verification
Thinking parallel lines have the same $q$	Parallel means same $k$ , not same $q$
Writing $k = \frac{x _{2} - x _{1}}{y _{2} - y _{1}}$	The slope is $\frac{Δ y}{Δ x}$ , not $\frac{Δ x}{Δ y}$
Confusing with direct proportion	Direct proportion: $y = k x$ (no $q$ ). Linear: $y = k x + q$

12. Interactive Exercises {#interactive-exercises}

Practice linear functions:

Find Slope and Y-intercept

Summary

Concept	Description
Linear function	$f (x) = k x + q$
Slope $k$	Rate of change; steepness and direction of the line
Y-intercept $q$	Where the line crosses the y-axis: $(0, q)$
Drawing	Find 2 points, plot them, draw a straight line
Parallel	Same slope $k$ , different $q$
Special case	$q = 0$ gives direct proportion $y = k x$

Introduction to Functions -- what is a function
Direct Proportion -- the special case $y = k x$
Functions and Graphs -- plotting and reading graphs
Domain of a Function -- allowed inputs
Range of a Function -- possible outputs
Summary of Functions -- quick reference card

Linear Function

Article Contents

1. What is a Linear Function? {#what-is-linear}

Relationship to Direct Proportion

2. Slope k {#slope-k}

Example

3. Y-intercept q {#y-intercept-q}

4. Drawing the Graph (Two-Point Method) {#drawing-graph}

Step 1: Find two points

Step 2: Calculate f(x)

Step 3: Plot and connect

Example: f(x)=2x+1

5. SVG: Comparing Linear Functions {#comparing-functions}

6. Special Cases {#special-cases}

7. Parallel Lines {#parallel-lines}

Example

8. Finding the Equation from Two Points {#equation-from-points}

Example

9. Intersection with Axes {#intersection-axes}

Y-axis intersection

X-axis intersection (zero of the function)

Example: f(x)=2x−6

10. Properties {#properties}

11. Common Mistakes {#common-mistakes}

12. Interactive Exercises {#interactive-exercises}

Summary

Related Articles

Step 2: Calculate $f (x)$

Example: $f (x) = 2 x + 1$

Example: $f (x) = 2 x - 6$