Last reviewed: 11 November 2026
Updated November 2026. These Grade 9 coordinates and graphs notes show how to plot points on the Cartesian plane, how to draw a straight line graph from an equation, and how to recognise parallel and perpendicular lines from their gradients. A point (3, 2) sits 3 units right of the origin and 2 units up. Two lines that never meet have the same gradient; two lines that meet at a right angle have gradients whose product is −1.
The examples are original practice for Kenyan junior school learners, not an official paper. They follow the Grade 9 Mathematics geometry sub-strand on coordinates and graphs in the KICD Grade Nine curriculum designs. Finding the equation of a line from two points is covered in the Grade 9 equation of a straight line notes. For the topic list, use the Grade 9 maths revision guide and the Grade 9 notes hub.
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Plotting Points and Drawing a Straight Line Graph
On the Cartesian plane the first number in a pair is the x-coordinate (across) and the second is the y-coordinate (up or down). Negative x is to the left of the origin; negative y is below it. To draw the graph of an equation such as y = 2x + 1, make a table of values, plot the points and join them with a straight line. The same table-and-plot method is used in OpenStax Elementary Algebra’s section on graphing linear equations.
Worked example: a table of values
Draw the graph of y = 2x + 1 for x from −1 to 2.
| x | −1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y = 2x + 1 | −1 | 1 | 3 | 5 |
Plot (−1, −1), (0, 1), (1, 3) and (2, 5). The points lie on one straight line that crosses the y-axis at (0, 1). That crossing is the intercept; the line rises 2 units for every 1 unit it moves right, so the gradient is 2.
Parallel Lines and Their Gradients
Parallel lines have the same gradient and never meet. The line y = 3x + 2 has gradient 3. Any line y = 3x + c, with a different intercept c, is parallel to it. The intercept changes where the line crosses the y-axis, but not its steepness. OpenStax Intermediate Algebra’s section on slope uses the same test for parallel lines.
Worked example
Show that y = 3x + 2 and y = 3x − 4 are parallel, and sketch how they sit.
Both equations have gradient 3, so the lines are parallel. The first crosses the y-axis at (0, 2); the second crosses at (0, −4). They stay 6 units apart measured vertically and never meet.
The line y = 2x + 5 is not parallel to them, because its gradient is 2, not 3.
Perpendicular Lines and Their Gradients
Two lines are perpendicular when they meet at a right angle. Their gradients m1 and m2 then satisfy m1 × m2 = −1. That means each gradient is the negative reciprocal of the other: if one line has gradient 2, a perpendicular line has gradient −1/2. If one has gradient −3, a perpendicular line has gradient 1/3.
Worked example
A path on a plan is the line y = 2x + 1. Write the equation of a perpendicular path that passes through (0, 3).
The given gradient is 2, so the perpendicular gradient is −1/2. A line through (0, 3) with that gradient is y = (−1/2)x + 3. Check: 2 × (−1/2) = −1, so the lines are perpendicular. They meet when 2x + 1 = (−1/2)x + 3. Multiply through by 2: 4x + 2 = −x + 6, so 5x = 4 and x = 4/5. The meeting point exists and the product of gradients confirms the right angle.
Original Practice Questions With Worked Answers
- Plot the points (3, 2), (−4, 1) and (0, −5). In which quadrant is (−4, 1)? (−4, 1) is in the second quadrant (left of the origin and above the x-axis).
- Complete the table for y = −x + 4 when x = 0, 1 and 2, then state the intercept. The points are (0, 4), (1, 3) and (2, 2). The line crosses the y-axis at (0, 4).
- Which of y = 5x − 1, y = 5x + 7 and y = −5x + 2 are parallel? The first two, because both have gradient 5. The third has gradient −5.
- A line has gradient 4. What is the gradient of a line perpendicular to it? −1/4, because 4 × (−1/4) = −1.
- Are y = (1/3)x + 2 and y = −3x + 1 perpendicular? Yes: (1/3) × (−3) = −1.
- A learner says two lines with gradients 2 and 1/2 are perpendicular. What is wrong? The product is 1, not −1. The second gradient should be −1/2.
Cover the answers and try each question. Grade 8 plotting practice is in the Grade 8 coordinates and graphs notes if you want a warmer start. For timed work, use the Grade 9 revision papers or the Grade 9 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
How do you draw a straight line graph from an equation?
Choose three or four values of x, calculate the matching y values, plot the points on the Cartesian plane and join them with a straight line. Extra points help you catch an arithmetic slip.
How can you tell that two lines are parallel?
They have the same gradient and different intercepts. y = 3x + 2 and y = 3x − 4 are parallel because both have gradient 3.
What is the rule for perpendicular gradients?
The product of the two gradients is −1. Each gradient is the negative reciprocal of the other, so a gradient of 2 pairs with −1/2.
Do horizontal and vertical lines fit these rules?
A horizontal line has gradient 0 and a vertical line has an undefined gradient. They are perpendicular to each other, but you cannot multiply by an undefined gradient, so treat that pair as a special case.