What is covered in Grade 9 Mathematics Graphs?

Interpret and draw linear graphs; x- and y-intercepts; gradient; determine equations from given graphs.

Is Grade 9 Graphs tested in Paper 1 or Paper 2?

Grade 9 Mathematics Graphs is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Graphs taught in Grade 9 Mathematics?

Graphs is taught in Term 2 of Grade 9 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 9 Mathematics Graphs on the CAPS curriculum?

Yes. Graphs is part of the official CAPS Mathematics curriculum for Grade 9, Term 2 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 9 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 9 Mathematics

Grade 9 · Term 2Mathematics

Graphs

We interpret and draw linear graphs using the gradient-intercept form, find x- and y-intercepts, and determine equations from graphs. We also explore the gradient as a rate of change and distinguish between different types of linear situations.

3.1 Properties of Linear Graphs 3.2 Finding Equations from Graphs

Weeks 5–6

3.1 Properties of Linear Graphs

Draw linear graphs from equations using a table of values or intercept method
Identify and calculate the x-intercept and y-intercept
Define gradient (slope) and calculate it from two points or the equation
Interpret gradient as rate of change in context

🌍

Real-World Connection

The gradient of a line is its 'steepness.' A wheelchair ramp must have a gentle gradient (1:12 — rise 1m for every 12m horizontal). A ski slope has a steep gradient. On a distance-time graph, gradient = speed. On a cost graph, gradient = price per unit. The steeper the line, the faster the rate of change.

The Cartesian plane: horizontal x-axis, vertical y-axis, origin at (0, 0). Quadrants I–IV.

A linear graph is a straight line. It shows a constant rate of change between y and x.

Gradient-Intercept Form

y = mx + c

m = gradient (slope), c = y-intercept (where the line crosses the y-axis)

Gradient Formula

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}

(x₁, y₁) and (x₂, y₂) are any two points on the line

Gradient and Slope Direction

Positive gradient (m > 0)

Negative gradient (m < 0)

Direction

Line rises left to right ↗

Line falls left to right ↘

Example

y = 2x + 1

y = −3x + 4

Special

m = 0: horizontal line

Undefined m: vertical line

💡 Tip

To find the x-intercept: let y = 0 and solve for x. To find the y-intercept: let x = 0 and solve for y (or just read off the 'c' value if in y = mx + c form).

Worked Examples

Worked Example

Drawing a linear graph and finding intercepts

Problem

y = 2x - 4

Worked Example

Gradient as rate of change in context

Problem

C = -8t + 100

Worked Example

Drawing a line using a table of values

Problem

y = -\frac{1}{2}x + 3

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

For

y = 3x + 6

, state the gradient and y-intercept.

L1 · Knowledge2 marks

Calculate the gradient of the line passing through

(1, 3)

and

(4, 9)

L2 · Routine Procedures4 marks

Find the x- and y-intercepts of

2x - 3y = 12

L2 · Routine Procedures4 marks

A mobile data plan charges R50 per month plus R0.20 per MB. Write the equation for cost

C

in terms of data

d

(MB). What does the gradient represent?

L3 · Complex Procedures5 marks

Show that the points

A(−1, −5)

B(1, 1)

, and

C(3, 7)

are collinear (lie on the same straight line).

L3 · Complex Procedures3 marks

The equation of a line is

y = mx + 2

and it passes through

(3, 8)

. Find

m

L4 · Problem Solving5 marks

Two lines:

y = 2x + 1

and

y = -x + 7

. Find their intersection point.

L4 · Problem Solving6 marks

A cyclist travels at 15 km/h. A runner starts at the same time but only at 5 km/h, but the cyclist starts 20 km behind the runner. Write equations for both distances from start and find when and where the cyclist overtakes the runner.

Week 7

3.2 Finding Equations from Graphs

Determine the equation of a line from a graph using the gradient and y-intercept
Determine the equation from two given points
Determine if a point lies on a given line

🌍

Real-World Connection

A scientist measures two data points from an experiment and wants to model the relationship. By reading the gradient and y-intercept from the graph, they write an equation that predicts future values. This is the essence of mathematical modelling — used in climate science, economics, and medicine.

Method: Finding the Equation of a Line

From a graph: read the y-intercept (c) directly. Calculate gradient m using two clear grid points.
Substitute m and c into y = mx + c.
From two points: calculate m using the gradient formula. Substitute m and one point into y = mx + c to find c.

ℹ️ Note

Special lines: y = c is a horizontal line with gradient 0 (e.g. y = 3). x = k is a vertical line with undefined gradient (e.g. x = −2). These cannot be written in y = mx + c form.

Worked Examples

Worked Example

Reading the equation directly from a graph

Problem

(0,\, 4)

Worked Example

Determining the equation from two points

Problem

A(-1, 7)

Worked Example

Testing if a point lies on a line

Problem

P(4, -3)

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

A line has gradient 3 and y-intercept −2. Write its equation.

L1 · Knowledge2 marks

Does the point

(2, 7)

lie on the line

y = 3x + 1

L2 · Routine Procedures4 marks

Find the equation of the line through

(0, 5)

and

(3, −1)

L2 · Routine Procedures4 marks

A line passes through

(1, 4)

and

(5, 12)

. Find its equation.

L3 · Complex Procedures5 marks

The equation of a straight line is

2x + 3y = 12

. (a) Find the x-intercept by setting

y = 0

. (b) Find the y-intercept by setting

x = 0

. (c) Calculate the gradient using the two intercepts. (d) Rewrite the equation in the form

y = mx + c

L3 · Complex Procedures4 marks

The graph of

y = mx + c

passes through

(−1, 8)

and is parallel to

y = 3x − 2

. Find

m

and

c

L4 · Problem Solving5 marks

Three points are

A(−3, 7)

B(0, k)

, and

C(3, 1)

. For what value of

k

are A, B, C collinear?

L4 · Problem Solving4 marks

A company's profit (R thousands) after

x

months follows

P = 3x - 6

. After how many months does it break even? What does the y-intercept mean in this context?

Algebraic Equations

Geometry of Straight Lines