Algebra · Patterns and Relations · Lesson 4
The Number Plane: Tables of Values and Straight-Line Graphs
A rule can be shown as words, as a table and as a picture. The picture is a graph on the number plane, and it lets you see a whole relation at a glance. In this lesson you will locate points using coordinates, build a table of values from a rule such as , plot the points, join them with a straight line, and read answers from your graph — the same skills used to read maps and fare charts.
By the end you will be able to
- ✔Name and plot points such as (1, 2) and (−3, 2) on the number plane, including negative coordinates.
- ✔Draw a table of values for a given rule over a given domain.
- ✔Plot the points from a table of values and join them to draw the graph of the relation.
- ✔Use a graph or a number plane to solve problems, such as reading values and finding locations.
Key words
- Number plane
- A flat grid made by two number lines (axes) crossing at right angles, used to locate points.
- x-axis and y-axis
- The horizontal number line is the x-axis; the vertical number line is the y-axis.
- Origin
- The point where the axes cross, O(0, 0).
- Coordinates
- An ordered pair (x, y) that gives a point's position: x across, then y up or down.
- Table of values
- A table that lists inputs (x) and the outputs (y) the rule gives, ready to plot as points.
Curriculum reference
Content standard: B7.2.1.1 Derive the rule for a set of points of a relation, draw a table of values to graph the relation in a number plane and make predictions about subsequent elements of the relation.
Indicator: Locate points on the number plane, draw a table of values of a given relation, draw graphs for given relations and use them to solve problems.
Starter
Finding your seat at the Accra Sports Stadium
Your ticket for the Black Stars match says Block C, Row 7, Seat 12. Two numbers together tell the steward exactly where you sit — not just the row, not just the seat, but both, in the right order. Maths uses the same idea to fix any point on a flat grid, and once points are fixed we can draw a rule as a picture and read answers straight off it.
1. Locating points on the number plane
The number plane is made of two number lines. The horizontal one is the x-axis and the vertical one is the y-axis. They cross at the origin, O(0, 0). To the right of O and above O the numbers are positive; to the left and below they are negative.
Every point has coordinates written as an ordered pair (x, y). The first number, x, tells you how far to move across (right if positive, left if negative). The second number, y, tells you how far to move up (positive) or down (negative).
▶ Always move ACROSS first (the x-coordinate), then UP or DOWN (the y-coordinate). Notice that negative numbers mean left or down.
2. Making a table of values from a rule
A rule such as tells you how to find y for any x. The set of x-values you are told to use is the domain. Put each x into the rule and write the y-value underneath.
- 1Draw a table with an x row and a y row. Write the domain in the x row: −2, −1, 0, 1, 2, 3.
- 2Substitute each x into the rule. x = −2: y = 2 × (−2) + 1 = −4 + 1 = −3.
- 3Carry on: x = −1 gives −1, x = 0 gives 1, x = 1 gives 3, x = 2 gives 5, x = 3 gives 7.
- 4Check the pattern: when x goes up by 1, y goes up by 2 every time. If one value breaks the pattern, recheck it.
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y = 2x + 1 | −3 | −1 | 1 | 3 | 5 | 7 |
3. Drawing the graph
Each column of the table is a point: (−2, −3), (−1, −1), (0, 1), (1, 3), (2, 5), (3, 7). Choose a sensible scale for each axis, plot every point with a small cross or dot, then join them with a ruler. For rules like y = 2x + 1 the points always lie on a straight line, so we call them linear relations.
▶ Each x in the table gives one point. Notice that all the points line up, so a ruler joins them in one straight line.
Rules can also make the line slope downwards. For y = −2x + 5 the table is x: 0, 1, 2, 3, 4 and y: 5, 3, 1, −1, −3. Here y goes down by 2 each time x goes up by 1, so the line falls from left to right.
▶ A negative multiplier of x makes the line slope downwards from left to right.
4. Using graphs to solve problems
Once a graph is drawn you can read values from it. To find y for a given x, go up (or down) from x to the line, then across to the y-axis. To find x for a given y, go across from y to the line, then down to the x-axis.
A taxi in Sunyani charges GH₵3 to start plus GH₵2 per kilometre, so fare = 2 × distance + 3. The table for 0 to 5 km is: 3, 5, 7, 9, 11, 13. From the graph, a fare of GH₵11 lines up with 4 km, and 2.5 km lines up with GH₵8. The graph also shows that the fare for 0 km is GH₵3 — the starting charge — where the line meets the y-axis.
| Distance (km) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Fare (GH₵) | 3 | 5 | 7 | 9 | 11 | 13 |
A number plane can also act as a map. If the school gate is at the origin, the library at (2, 7) and the canteen at (−3, 2), you can describe a route by moving across and up, and you can count how far apart places are along the grid lines.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Plotting points with negative coordinates
Show the solution
- Step 1(1, 2): from O move 1 unit right, then 2 units up.
- Step 2(−3, 2): from O move 3 units left, then 2 units up.
- Step 3Both have y = 2, so they are on the same horizontal line.
- Step 4Distance = 1 − (−3) = 1 + 3 = 4 units.
Answer: They are 4 units apart on the line y = 2.
Example 2
Table of values for y = 2x + 1
Show the solution
- Step 1x = −2: 2(−2) + 1 = −3; x = −1: 2(−1) + 1 = −1.
- Step 2x = 0: 0 + 1 = 1; x = 1: 2 + 1 = 3.
- Step 3x = 2: 4 + 1 = 5; x = 3: 6 + 1 = 7.
Answer: y: −3, −1, 1, 3, 5, 7
Example 3
A table for a decreasing rule
Show the solution
- Step 1x = −2: 4 − (−2) = 6; x = −1: 4 − (−1) = 5.
- Step 2x = 0: 4; x = 1: 3; x = 2: 2; x = 3: 1.
- Step 3Check: y goes down by 1 each time x goes up by 1 ✓
Answer: y: 6, 5, 4, 3, 2, 1
Example 4
Finding the rule from a graph's table
▶ The points fall 2 units for every 1 unit across.
Show the solution
- Step 1When x goes up by 1, y goes down by 2, so the rule has −2x.
- Step 2When x = 0, y = 5, so the rule is y = −2x + 5.
- Step 3Check: x = 3 gives −6 + 5 = −1 ✓
- Step 4x = 6: y = −12 + 5 = −7.
Answer: y = −2x + 5; when x = 6, y = −7
Example 5
Reading a fare graph
Show the solution
- Step 1Table: 0 → 3, 1 → 5, 2 → 7, 3 → 9, 4 → 11, 5 → 13.
- Step 2(i) Go up from 4 km to the line and across: GH₵11.
- Step 3(ii) Go across from GH₵13 to the line and down: 5 km.
- Step 4Check: 2 × 5 + 3 = 13 ✓
Answer: (i) GH₵11 (ii) 5 km
Summary
- Coordinates (x, y): move across by x first (right +, left −), then up or down by y (up +, down −).
- The origin O(0, 0) is where the x-axis and y-axis cross.
- A table of values is made by substituting each x in the domain into the rule; use brackets for negative x.
- Plot each (x, y) pair and join with a ruler — rules like y = 2x + 1 give straight lines.
- A positive multiplier of x makes the line rise from left to right; a negative one makes it fall.
- Read a graph by going from one axis to the line, then across (or down) to the other axis.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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