10-Week Mathematics Catch-Up Programme (CAPS)

Week 3 – Day 3

Gradient of a Straight Line

Today we build on the work from yesterday. You already know that the gradient tells us how steep a line is. Today you'll learn how to calculate the gradient from two points, recognise parallel and perpendicular lines, and interpret what a gradient means in real-life contexts.

This lesson is particularly important because gradient is tested repeatedly in Grades 10–12 and appears in coordinate geometry, functions, calculus and even physics. Duration: 1 Hour

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

Learning Outcomes

By the end of this lesson you should be able to:

• Understand what the gradient (slope) represents.
• Calculate the gradient from two points.
• Recognise positive, negative, zero and undefined gradients.
• Identify parallel and perpendicular lines.
• Interpret gradients in real-life situations.

Part A – Study Guide

1. What is a Gradient?

The gradient (also called the slope) tells us how steep a line is.

It tells us how much the graph goes up or down as we move from left to right.

Positive Gradient

A line rising from left to right.

Example:

$y=2x+1$

Gradient = 2

Negative Gradient

A line falling from left to right.

Example:

$y=-3x+4$

Gradient = -3

Zero Gradient

A horizontal line.

Example:

$y=5$

Gradient = 0

Undefined Gradient

A vertical line.

Example:

$x=4$

The gradient is undefined because there is no horizontal movement.

2. Calculating Gradient from Two Points

The formula is:

$\boxed{\text{Gradient}=\frac{\text{Change in }y}{\text{Change in }x}}$

or

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

where:

• (m) = gradient
• ((x_1,y_1)) = first point
• ((x_2,y_2)) = second point

Example 1

Find the gradient between

A(2,3)

and

B(6,11)

Step 1: Change in (y)

$11-3=8$

Step 2: Change in (x)

$6-2=4$

Step 3:

$m=\frac84=2$

Answer:

Gradient = 2

Example 2

Find the gradient between

A(-2,4)

and

B(3,-6)

$m=\frac{-6-4}{3-(-2)}$

$=\frac{-10}{5}$

$=-2$

Answer:

Gradient = -2

3. Parallel Lines

Parallel lines always have the same gradient.

Example:

$y=3x+1$

and

$y=3x-5$

Both have gradient 3.

Therefore they are parallel.

4. Perpendicular Lines (Introduction)

Perpendicular lines meet at 90°.

At this stage, simply recognise that:

• Horizontal lines are perpendicular to vertical lines.
• Later (Grade 10/11), you'll learn the numerical rule:

$m_1 \times m_2=-1$

5. Real-Life Meaning of Gradient

Example 1

Distance-Time Graph

Gradient = speed.

A steeper line means travelling faster.

Example 2

Cost Graph

If

$y=12x+20$

Gradient = 12

This means the cost increases by R12 for every extra kilometre.

Common Mistakes

Mistake 1

Subtracting the coordinates in different orders.

If you calculate:

$y_2-y_1$

then you must calculate:

$x_2-x_1$

Use the same order for both.

Mistake 2

Mixing up x and y values.

Always write the points clearly first.

Mistake 3

Forgetting to simplify the fraction.

Example:

$\frac{8}{4}=2$

Part B – Worked Examples

Example 1

Find the gradient between

(1,2)

and

(5,10)

$m=\frac{10-2}{5-1} =\frac84 =2$

Example 2

Find the gradient between

(2,5)

and

(6,5)

$m=\frac{5-5}{6-2} =0$

A horizontal line.

Example 3

Find the gradient between

(4,2)

and

(4,9)

The change in (x) is 0.

Division by zero is impossible.

Gradient is undefined.

Example 4

Are these lines parallel?

$y=5x+7$

$y=5x-4$

Both have gradient 5.

Yes, they are parallel.

Example 5

A car travels 180 km in 3 hours.

Gradient of the distance-time graph:

$180\div3=60$

The gradient represents 60 km/h.

Part C – Practice Questions

Section A – Identify the Type of Gradient

State whether each line has a positive, negative, zero, or undefined gradient.

1. $y=4x+2$

2. $y=-2x+5$

3. $y=7$

4. $x=-3$

5. $y=x-8$

6. $y=-6x+1$

7. $y=0$

8. $x=10$

9. $y=3x-2$

10. $y=-x+6$

Section B – Find the Gradient

11. A(1,2), B(5,10)

12. A(3,7), B(7,15)

13. A(-2,4), B(2,8)

14. A(0,5), B(4,-3)

15. A(6,1), B(10,9)

16. A(2,5), B(8,5)

17. A(4,-2), B(4,7)

18. A(-3,-1), B(1,7)

19. A(5,3), B(9,-5)

20. A(-4,6), B(2,0)

Section C – Parallel Lines

State whether the following pairs of lines are parallel.

$y=2x+5$

and

$y=2x-8$

$y=4x+1$

and

$y=3x+1$

$y=-5x+2$

and

$y=-5x-9$

$y=x+7$

and

$y=2x+7$

$y=-3x+8$

and

$y=-3x-1$

Section D – Real-Life Interpretation

26. A taxi fare is given by

$y=18x+30$

What does the gradient represent?

27. A cyclist travels 90 km in 3 hours.

What is the gradient of the distance-time graph?

28. A graph has gradient 0.

Describe what the graph looks like.

29. A graph has an undefined gradient.

Describe what the graph looks like.

30. Which graph is steeper?

$y=6x+2$

or

$y=3x+2$

Explain.

Challenge Questions

31. Find the gradient between

(-5,-2)

and

(3,14)

32. A line passes through

(2,4)

and

(8,16)

Find the gradient.

33. Explain why the following lines are parallel:

$y=7x+3$

$y=7x-12$

34. A mountain hiking trail rises 500 metres over a horizontal distance of 2 km.

Calculate the gradient (rise/run).

(Convert 2 km to metres first.)

35. A learner says:

"All lines with a positive gradient are parallel."

Is the learner correct? Explain your answer.

Answers

Section A

1. Positive

2. Negative

3. Zero

4. Undefined

5. Positive

6. Negative

7. Zero

8. Undefined

9. Positive

10. Negative

Section B

$m=\frac{10-2}{5-1}=\frac84=2$

$m=\frac{15-7}{7-3}=\frac84=2$

$m=\frac{8-4}{2-(-2)}=\frac44=1$

$m=\frac{-3-5}{4-0}=\frac{-8}{4}=-2$

$m=\frac{9-1}{10-6}=\frac84=2$

$m=0$

Undefined

$m=\frac{7-(-1)}{1-(-3)}=\frac84=2$

$m=\frac{-5-3}{9-5}=\frac{-8}{4}=-2$

$m=\frac{0-6}{2-(-4)}=\frac{-6}{6}=-1$

Section C

21. Yes

22. No

23. Yes

24. No

25. Yes

Section D

26. The fare increases by R18 per kilometre.

$90\div3=30$

30 km/h

28. A horizontal line.

29. A vertical line.

30. (y=6x+2) is steeper because its gradient (6) is greater than 3.

Challenge Answers

$m=\frac{14-(-2)}{3-(-5)}=\frac{16}{8}=2$

$m=\frac{16-4}{8-2}=\frac{12}{6}=2$

33. Both equations have the same gradient (7), so the lines are parallel.

Convert 2 km to metres:

$2\text{ km}=2000\text{ m}$

Gradient:

$\frac{500}{2000}=\frac14=0.25$

35. No. Two lines are parallel only if they have exactly the same gradient. For example, (y=2x+1) and (y=5x+1) both have positive gradients, but they are not parallel because their gradients are different.

Parent's Notes

By the end of today's lesson, your student should be able to:

• Identify positive, negative, zero and undefined gradients.
• Calculate the gradient from two coordinates using the formula.
• Recognise parallel lines from their equations.
• Interpret gradients in practical situations such as speed and cost.

Tip: Encourage them to memorise the gradient formula:

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

It will be used throughout Grades 10–12 in coordinate geometry, functions and calculus.