Linear functions and their graphs

IB Mathematics AI SL· Topic 2.4· 5 min read

1. Definition and Key Features★☆☆☆☆⏱ 10 min

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📘 Definition

Linear Function

$f (x) = m x + c$

A first-degree function with a constant rate of change, that graphs as a straight line on the $x y$ -plane.

Example:

$f (x) = 2 x + 5$ is linear; $f (x) = x^{2} - 1$ is non-linear

In the standard gradient-intercept form $y = m x + c$ , $m$ is the gradient (measures steepness and direction) and $c$ is the $y$ -coordinate of the $y$ -intercept where the line crosses the $y$ -axis.

📐 Worked Example

Identify the gradient and $y$ -intercept of the line $2 y - 4 x = 12$

1
Rearrange the equation into gradient-intercept form $y = m x + c$ :
2
$2y = 4x + 12$
3
Divide all terms by 2 to isolate $y$ :
4
$y = 2x + 6$
5
Compare to $y = m x + c$ : gradient $m = 2$ , $y$ -intercept is $(0, 6)$

2. Finding the Equation of a Line★★☆☆☆⏱ 15 min

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To find the equation of a line, you need two pieces of information: either the gradient and one point on the line, or two points that lie on the line. The gradient between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is:

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

Once you have the gradient, use the point-gradient formula to find the full equation, then rearrange to gradient-intercept form:

y - y_1 = m(x - x_1)

📐 Worked Example

Find the equation of the line passing through $(2, 5)$ and $(4, 9)$

1
Calculate the gradient from the two points:
2
$m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2$
3
Substitute $m = 2$ and point $(2, 5)$ into point-gradient formula:
4
$y - 5 = 2(x - 2)$
5
Expand and rearrange to gradient-intercept form:
6
$y - 5 = 2x - 4 \implies y = 2x + 1$

3. Sketching and Contextual Interpretation★★☆☆☆⏱ 15 min

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Linear functions are very commonly used to model real-world scenarios. In context, the gradient represents a constant rate of change, and the $y$ -intercept represents an initial value when the input is 0. To sketch a line, you only need two distinct points.

📐 Worked Example

A coffee shop charges a fixed \ $5 r e n t a l f ee f or am ee t in g r oo m pl u s$ 2 per person for drinks. Write the linear function for total cost $C$ for $p$ people, and interpret the gradient and intercept.

1
Identify the fixed initial fee as the intercept, and the cost per person as the gradient: $c = 5$ , $m = 2$
2
$C = 2p + 5$
3
Interpretation: The intercept of 5 means you pay \ $5 e v e ni f n o p eo pl e a tt e n d, an d t h e g r a d i e n t o f 2 m e an s t o t a l cos t in cr e a ses b y$ 2 for each additional person.
4
To sketch, use two points: when $p = 0$ , $C = 5$ → $(0, 5)$ ; when $p = 10$ , $C = 25$ → $(10, 25)$ . Draw a straight line between these points for $p \geq 0$ (number of people can't be negative).

4. Intersection of Two Lines★★★☆☆⏱ 15 min

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The intersection of two lines is the point $(x, y)$ that satisfies both equations. In context, this often represents a break-even point or equilibrium value. You can find the intersection algebraically or graphically.

📐 Worked Example

Find the intersection of $y = 3 x - 4$ and $y = - x + 8$

1
Set the expressions for $y$ equal to each other, since they are equal at the intersection:
2
$3x - 4 = -x + 8$
3
Solve for $x$ :
4
$4x = 12 \implies x = 3$
5
Substitute $x = 3$ back into one equation to find $y$ :
6
$y = -(3) + 8 = 5$
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The intersection point is $(3, 5)$

✓ Quick check

Test your understanding:

What is the intersection of $y = 2 x + 1$ and $y = x + 4$ ?
- $(2, 5)$
- $(3, 7)$
- $(1, 3)$
- $(4, 8)$

5. Common Pitfalls

Wrong move:

Rearranging $a x + b y = c$ gives the wrong sign for the gradient $m$

Why:

Forgetting to change the sign of $a x$ when moving it to the other side of the equation

Correct move:

Rearrange step-by-step: move $a x$ to the right-hand side first, then divide all terms by $b$ to isolate $y$

Wrong move:

Calculating gradient as $\frac{x _{2} - x _{1}}{y _{2} - y _{1}}$ instead of rise over run

Why:

Mixing up the order of $x$ and $y$ differences in the gradient formula

Correct move:

Always remember gradient = change in $y$ (rise) divided by change in $x$ (run)

Wrong move:

Drawing the full infinite line across negative axes in contextual problems

Why:

Forgetting that quantities like distance or number of people can't be negative

Correct move:

Only draw the line for the valid range of the input variable, and label your domain

Wrong move:

Stopping after finding $x$ when asked for the intersection point

Why:

Forgetting that an intersection is a 2D point with both $x$ and $y$ coordinates

Correct move:

Always substitute your $x$ value back into one of the equations to find the corresponding $y$ value

6. Quick Reference Cheatsheet

Concept	Key Formula/Rule
Gradient between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$	$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Gradient-intercept form	$y = m x + c$ , $m$ =gradient, $c$ = $y$ -intercept
Point-gradient form	$y - y_{1} = m (x - x_{1})$
Find intersection of two lines	Set equal, solve for $x$ , find $y$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Find equation of line through two points
2023 · 1
Interpret linear graph in context
2021 · 2
Calculate intersection of two lines

Going deeper

syllabusIB AI SL Official SyllabusCheck full topic 2 requirements

What's Next

Linear functions are the foundational building block for almost all other topics in IB AI SL. Mastering these core skills will make working with linear regression, systems of equations, piecewise functions, and even non-linear functions much easier, because almost all non-linear relationships can be approximated by linear functions around a point. You will regularly use linear functions to model real-world data in statistics, solve optimization problems, and interpret relationships between variables across the entire course. Next, you will build on these skills to study more complex functions starting with quadratic functions, the next core topic in the functions unit.