DISCRETE MATHEMATICS · TEXTBOOK LESSON
Graphs show relationships visually. Coordinates locate points, gradients describe rates of change, and equations connect algebraic rules to geometric shapes.
Learning objectives
By the end of this lesson, you should be able to:
- Plot and read coordinates accurately.
- Interpret axes, scales and units.
- Calculate gradient and identify intercepts.
- Connect equations with the shape and position of graphs.
1. Key vocabulary and definition
Definition
For a straight line y = mx + c, m is the gradient and c is the y-intercept. Gradient = change in y / change in x.
coordinatex-axisy-axisgradientinterceptfunction
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- Coordinates are written (x,y): move horizontally first, then vertically.
- Positive gradient rises from left to right; negative gradient falls.
- Parallel non-vertical lines have equal gradients.
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Check the scale and units on both axes.
2
For gradient, choose two clear points on the line.
3
Compute change in y and change in x in the same direction.
4
Divide change in y by change in x.
5
Use the intercept and gradient to verify the equation.
4. Worked example
Worked example
Find the equation of a line with gradient 3 and y-intercept -2.
1
Use the straight-line form y = mx + c.
2
Gradient m = 3.
3
y-intercept c = -2.
4
Therefore the equation is y = 3x - 2.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- Plot the point (3,-2).
- Find the gradient between (1,4) and (5,12).
- State the y-intercept of y=4x+7.
- Find the equation of a line with gradient -2 and intercept 5.
- Are y=3x+1 and y=3x-8 parallel?
Show answers
- Point 3 right and 2 down from the origin
- 2
- 7
- y = -2x + 5
- Yes
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.