10-Week Mathematics Catch-Up Programme (CAPS)

Week 3 – Day 2

Tables of Values and Straight-Line Graphs

Today we move into one of the most important topics in high school mathematics: functions and straight-line graphs.

Many Grade 11 learners struggle because they try to memorise graphs instead of understanding the connection between an equation, a table of values, and the graph itself. This lesson builds that understanding step by step.

Duration: 1 Hour

Grade Level: Grade 9 Foundation (Preparing for Grade 10 & 11 Functions)

Learning Outcomes

By the end of today's lesson you should be able to:

Understand what a function is.
Complete a table of values.
Plot points from a table.
Draw a straight-line graph accurately.
Identify the gradient and y-intercept from a graph.

Part A – Study Guide

1. What is a Function?

A function is a rule that connects an input (x-value) to an output (y-value).

For example:

$y = x + 2$

This means:

Take the x-value and add 2 to get the y-value.

Example

$x=3$

then

$y=3+2=5$

So one point on the graph is

$(3,5)$

2. Making a Table of Values

Suppose

$y=x+2$

Choose several x-values.

x	-2	-1	0	1	2
y	0	1	2	3	4

These coordinate pairs are:

$(-2,0);(-1,1);(0,2);(1,3);(2,4)$

3. Plotting the Points

Plot each coordinate on graph paper.

Then use a ruler to join the points.

You should get a straight line.

4. The y-Intercept

The y-intercept is where the graph crosses the y-axis.

$y=x+2$

when

$x=0$

$y=2$

The graph crosses the y-axis at

$(0,2)$

5. The Gradient (Slope)

The gradient tells us how steep a line is.

For

$y=x+2$

every time x increases by 1, y also increases by 1.

Gradient = 1

For

$y=2x+1$

every increase of 1 in x causes y to increase by 2.

Gradient = 2

For

$y=-x+4$

every increase of 1 in x causes y to decrease by 1.

Gradient = -1

6. Understanding the Equation

Every straight-line equation can be written as:

$y=mx+c$

where:

(m) = gradient (slope)
(c) = y-intercept

Example:

$y=3x-5$

Gradient = 3

y-intercept = -5

Common Mistakes

Mistake 1

Substituting incorrectly.

Example:

$y=2x+1$

If (x=4):

$y=2(4)+1=9$

Not (2+4+1=7).

Mistake 2

Joining points freehand.

Always use a ruler for straight-line graphs.

Mistake 3

Forgetting to label the axes.

Always label:

Horizontal axis: x
Vertical axis: y

Part B – Worked Examples

Example 1

Complete the table for

$y=x+3$

x	-2	-1	0	1	2
y	1	2	3	4	5

Example 2

Complete the table for

$y=2x$

x	-2	-1	0	1	2
y	-4	-2	0	2	4

Example 3

Complete the table for

$y=2x+1$

x	-2	-1	0	1	2
y	-3	-1	1	3	5

Example 4

State the gradient and y-intercept of

$y=4x-2$

Gradient:

$4$

y-intercept:

$-2$

Example 5

State the gradient and y-intercept of

$y=-3x+5$

Gradient:

$-3$

y-intercept:

$5$

Part C – Practice Questions

Section A – Complete the Tables

1.

$y=x+1$

x	-2	-1	0	1	2
y

2.

$y=2x+2$

x	-2	-1	0	1	2
y

3.

$y=-x+4$

x	-2	-1	0	1	2
y

4.

$y=3x$

x	-2	-1	0	1	2
y

5.

$y=2x-3$

x	-2	-1	0	1	2
y

Section B – Plot the Graphs

Using graph paper:

Plot the graph of

$y=x+1$

Plot the graph of

$y=2x$

Plot the graph of

$y=-x+4$

Plot the graph of

$y=3x-2$

Plot the graph of

$y=2x-3$

Section C – Gradient and y-Intercept

State the gradient and y-intercept.

$y=5x+2$

$y=4x-3$

$y=-2x+6$

$y=x-8$

$y=-5x+1$

Section D – Mixed Questions

What is the y-intercept of

$y=6x+4$

What is the gradient of

$y=-4x+7$

$y=3x+2$

find y when

$x=5$

$y=-2x+6$

find y when

$x=-3$

Which graph is steeper?

$y=5x$

$y=2x$

Explain your answer.

Challenge Questions

Complete the table for

$y=-2x+1$

using x-values:

-3, -2, -1, 0, 1, 2, 3.

Which equation has the larger gradient?

$y=7x-1$

$y=4x+8$

Write the equation of a straight line with:

Gradient = 3
y-intercept = -2

Write the equation of a straight line with:

Gradient = -1
y-intercept = 5

A taxi charges a fixed fee of R15 plus R12 per kilometre.

Let:

(x) = kilometres travelled
(y) = total cost

a) Write an equation for the cost.

b) Complete a table for (x = 0, 1, 2, 3, 4).

c) Plot the graph on graph paper.

Answers

Section A

1. (y=x+1)

x	-2	-1	0	1	2
y	-1	0	1	2	3

2. (y=2x+2)

x	-2	-1	0	1	2
y	-2	0	2	4	6

3. (y=-x+4)

x	-2	-1	0	1	2
y	6	5	4	3	2

4. (y=3x)

x	-2	-1	0	1	2
y	-6	-3	0	3	6

5. (y=2x-3)

x	-2	-1	0	1	2
y	-7	-5	-3	-1	1

Section C

Gradient = 5, y-intercept = 2
Gradient = 4, y-intercept = -3
Gradient = -2, y-intercept = 6
Gradient = 1, y-intercept = -8
Gradient = -5, y-intercept = 1

Section D

4
-4

$y=3(5)+2=17$

$y=-2(-3)+6=12$

(y=5x) is steeper because its gradient (5) is greater than 2.

Challenge Answers

x	-3	-2	-1	0	1	2	3
y	7	5	3	1	-1	-3	-5

(y=7x-1), because 7 > 4.

$y=3x-2$

$y=-x+5$

$y=12x+15$

x	0	1	2	3	4
y	15	27	39	51	63

c) Plot the five points:

$(0,15),\ (1,27),\ (2,39),\ (3,51),\ (4,63)$

and join them with a straight line.

Parent's Notes

This is the first lesson where graph paper is essential. Encourage your sstudent to:

Use a ruler to draw the axes and the graph.
Choose a sensible scale (for example, 1 square = 1 unit).
Plot each point carefully before joining them.
Label both axes and give the graph a title.

If they can confidently move between an equation, a table of values and a graph, they will have built one of the strongest foundations for the Grade 10–11 Functions section.