Practice Maths

Cartesian Plane: Graphing

From Patterns to Graphs

Key Terms

linear relationship
a relationship where the y-value changes by the same amount each time x increases by 1; produces a straight-line graph
non-linear relationship
a relationship where the y-value does not change by a constant amount; the graph is a curve
table of values
a table showing matched pairs of x and y values that satisfy a rule
coordinate pair
two numbers written as (x, y) that identify a point's exact position on the Cartesian plane

1. Points on a Straight Line (Linear)

We can turn patterns into pictures! If you have a number pattern, you can:

  1. Create a table of values.
  2. Turn them into coordinates: (x, y).
  3. Plot them on a Cartesian plane.

Example: A catering company provides 2 pizzas for every 1 table of guests.

Tables (x)1234
Pizzas (y)2468

Coordinates: (1, 2), (2, 4), (3, 6), (4, 8).

If you plot these, they form a straight line. We call this a Linear Relationship.

2. Non-Linear Patterns

Not all patterns make straight lines. Some patterns grow very quickly (like square numbers: 1, 4, 9, 16...).

If you plot these points, the line will curve. We call this a Non-Linear Relationship.

Hot Tip
  • Linear: The graph is a straight ruler line. The pattern adds (or subtracts) the same amount every time.
  • Non-Linear: The graph is a curve. The pattern changes by different amounts each time.

Worked Example

Graph y = 2x − 1 using a table of values.

Step 1 — Make a table for x = 0, 1, and 2.

x012
y = 2x − 1

Step 2 — Substitute each x-value into the rule to find y:
x = 0: y = 2(0) − 1 = −1    x = 1: y = 2(1) − 1 = 1    x = 2: y = 2(2) − 1 = 3

Step 3 — Plot the coordinate pairs (0, −1), (1, 1), and (2, 3) on the Cartesian plane, then draw a straight line through them. Because the y-values increase by the same amount (+2) each time, this is a linear relationship.

From Dots to Lines: Connecting the Points

In the previous lesson, you learned to plot individual points. Now we ask: what happens when you plot many points from the same rule and connect them? You get a graph — and the shape of that graph tells you something important about the relationship.

Think about tracking a plant's height each week. Each measurement is a separate dot on the graph. When you connect those dots with a line, you can see the trend — is it growing faster or slower? Is it accelerating or steady? The graph makes patterns visible that are hard to see in a table alone.

Linear Relationships: The Straight Line

A relationship is linear if the y-value changes by the same amount every time x increases by 1. This constant change produces a perfectly straight line on the graph.

From the pizza catering example in the Key Ideas tab (y = 2x), every extra table of guests requires exactly 2 more pizzas. The graph rises at a constant rate — same steepness all the way up. This is a classic linear relationship.

  • How to check: find the difference between consecutive y-values. If the differences are all equal, it's linear.
  • Example: y values of 3, 5, 7, 9 — differences are 2, 2, 2. All equal → linear!

Non-Linear Relationships: When the Line Curves

Not all patterns are linear. If you invest money and it earns compound interest, it grows faster and faster over time — that's non-linear. Square numbers (1, 4, 9, 16…) grow by increasing amounts, so their graph curves upward. The differences aren't equal (they go up by 3, 5, 7, 9…), which signals a non-linear relationship.

Remember: If the differences between consecutive y-values are all the same → straight line (linear). If the differences keep getting bigger or smaller → curved line (non-linear). Check the differences first before deciding which type you have.

Discrete vs Continuous Graphs

Should you join the dots with a solid line, or just leave the dots separate? It depends on whether the data is continuous or discrete:

  • Discrete: only specific values make sense (e.g., number of people — you can't have 2.5 people). Leave the points separate.
  • Continuous: any value in between is possible (e.g., temperature, distance, time). Draw a solid line through the points.

Reading Graphs: What the Intercepts Tell You

The point where a line crosses the y-axis (when x = 0) is called the y-intercept. It tells you the starting value — for example, if x = time and y = water in a tank, the y-intercept tells you how much water was there at the start. Understanding intercepts helps you make real-world sense of any graph you encounter.

Common Mistake: Drawing the line only between the first and last plotted points, without extending it. For continuous relationships, the line should extend in both directions (unless there's a real-world limit, like time can't be negative). Extending the line lets you make predictions beyond your data.

Practice Questions

  1. Reading Points from a Graph Fluency

    Identify the coordinates for points A, B, C, and D on the graph below.

    x y 2 4 6 -2 -4 -6 2 4 6 -2 -4 -6 A B C D
    1. Write the ordered pair for A.
    2. Write the ordered pair for B.
    3. Write the ordered pair for C.
    4. Write the ordered pair for D.

  2. Table to Coordinates Fluency

    Look at this table of values describing a pattern:

    x123
    y357
    1. Write these as a set of ordered pairs: (1, 3), ...
    2. Is this pattern likely to form a straight line or a curve?

  3. Graphing a Linear Pattern Understanding

    The pattern from Question 2 is plotted below. Does it pass through the point (0, 1)?

    2 4 2 4 6

  4. The Fencing Problem Understanding

    A builder is making a fence. He needs 4 posts for 1 section, but because they share a post, he only needs 7 posts for 2 sections.

    Sections (x)123
    Posts (y)4710
    1. Write the three coordinates.
    2. If you plotted these, would it be linear (straight) or non-linear (curved)?

  5. Currency Exchange (Linear) Fluency

    You are traveling to Japan. The exchange rate is roughly 1 AUD = 100 Yen.

    1. If you exchange $10, how many Yen do you get?
    2. If you exchange $20, how many Yen do you get?
    3. This creates coordinates (10, 1000) and (20, 2000). Is this graph linear?

  6. Comparing Steepness Understanding

    Imagine the exchange rate changed.

    • Rate A: 1 AUD = 100 Yen
    • Rate B: 1 AUD = 50 Yen

    If you graphed both lines on the same chart, which line would be steeper (go up faster)?

  7. Area of Squares (Non-Linear) Understanding

    The area of a square is calculated by side × side.

    Side length (x)1234
    Area (y)14916
    1. Plotting these points (1,1), (2,4), (3,9)... creates a specific shape. Is it a straight line?
    2. What is this type of relationship called?

  8. Visual Identification Fluency

    Look at the two graph shapes below:

    Graph A Graph B
    1. Which graph represents a linear relationship?
    2. Which graph represents a non-linear relationship?

  9. Interpreting Intercepts Understanding

    A graph describes the water level in a bath. The line starts at (0, 0) and goes up.

    1. What does the point (0, 0) mean in this real-life situation?
    2. If the line started at (0, 5), what would that mean?

  10. Predicting from the Graph Problem Solving

    A linear graph follows the rule y = 2x. We have plotted (1, 2), (2, 4), and (3, 6).

    Without calculating, if you followed the straight line with a ruler, what would the y-value be when x = 5?

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