Practice Maths

Plotting Linear Relationships

Key Ideas

Key terms: table of values, linear relationship, gradient, y-intercept, rise, run

• A linear relationship produces a straight line when graphed.
• To graph, create a table of values by substituting x values into the rule.
• Gradient = rise ÷ run = change in y ÷ change in x.
• A positive gradient slopes upward (left to right); negative slopes downward; zero is horizontal.
• The y-intercept is where the line crosses the y-axis (when x = 0).

Example: y = 2x − 1

Table of values for x = −2, −1, 0, 1, 2:

x: −2, −1, 0, 1, 2
y: −5, −3, −1, 1, 3

Gradient = 2 (rises 2 for every 1 across); y-intercept = −1

Hot Tip Always use at least 3 points when graphing. If they don’t all lie on one straight line, recheck your substitutions — you have likely made an arithmetic error.

The Cartesian Plane and Tables of Values

To plot a linear relationship, start by making a table of values. Choose several values of x (the input), substitute each into the rule to find y (the output), then write the (x, y) pairs as coordinates.

Example: Rule is y = 2x + 1. Choose x = 0, 1, 2, 3, 4.
• x = 0: y = 2(0) + 1 = 1 → point (0, 1)
• x = 1: y = 2(1) + 1 = 3 → point (1, 3)
• x = 2: y = 2(2) + 1 = 5 → point (2, 5)
• x = 3: y = 2(3) + 1 = 7 → point (3, 7)

Plot these on a Cartesian plane (horizontal x-axis, vertical y-axis) and connect them with a ruler. The result is a perfectly straight line.

Recognising a Linear Relationship

A relationship is linear if the points you plot lie on a straight line. This happens when y changes by a constant amount each time x increases by 1. In the example above, y goes up by 2 each time x goes up by 1 — this constant increase is the key feature of linearity.

If the y-values increase by different amounts each step, the relationship is non-linear (it might be a curve). For instance, y = x2 gives y values 0, 1, 4, 9, 16 — these increase by 1, 3, 5, 7 (not constant), so it is not linear.

Direct Proportion vs Non-Direct Proportion

Direct proportion (also called direct variation) means y = kx for some constant k. The graph is a straight line through the origin (0, 0). Doubling x doubles y.

Example: Buying apples at $2 each. Cost = 2 × number. The graph passes through (0, 0).

Non-direct proportion: the graph is still a straight line but does NOT pass through the origin. There is an extra constant added. Example: A phone plan costs $15/month plus $0.10 per text. Cost = 0.10 × texts + 15. When texts = 0, cost = $15, not $0. The starting point is not the origin.

Reading Information from a Graph

Once a linear relationship is graphed, you can read off any value. Find x on the horizontal axis, go straight up to the line, then across to the y-axis to read the y-value. Or work in reverse — find a y-value and read across to find x.

Example: A pizza shop charges $8 base price plus $3 per topping. Rule: Cost = 3 × toppings + 8. For 4 toppings: cost = $20. For 7 toppings: cost = $29. Plot this and read off costs for any number of toppings.

Why Linear Relationships Matter

Linear relationships are the foundation of algebra. They appear in science (distance = speed × time), finance (earnings = hourly rate × hours), cooking (cost per serving), and engineering. Understanding how to plot them and read from the graph is a skill you will use constantly in maths from here on.

Key tip: Always use at least three points when plotting a linear graph. Two points define a line, but the third point checks that you haven't made an arithmetic error. If all three points don't lie on the same straight line, one of your calculations is wrong — go back and check each substitution carefully.

Mastery Practice

  1. Complete a table of values for each rule using x = −2, −1, 0, 1, 2. Fluency

    1. y = x + 3
    2. y = 2x
    3. y = −x + 4
    4. y = 3x − 2
    5. y = −2x + 1
    6. y = x − 5
    7. y = 4x + 1
    8. y = −3x − 1

  2. State the gradient and y-intercept for each linear rule. Fluency

    1. y = 3x + 5
    2. y = −2x + 7
    3. y = x − 4
    4. y = −x
    5. y = 5x
    6. y = −4x − 3
    7. y = 2x + 0
    8. y = ½x + 2
    9. y = −3x + 6
    10. y = 10 − x

  3. Calculate the gradient between each pair of points. Fluency

    1. (0, 0) and (3, 6)
    2. (1, 5) and (3, 9)
    3. (0, 4) and (2, 0)
    4. (−1, 3) and (2, 9)
    5. (2, 8) and (5, 8)
    6. (0, 0) and (4, −12)
    7. (−2, −5) and (3, 10)
    8. (1, 7) and (4, 1)
    9. (−3, 2) and (1, 10)
    10. (5, −1) and (10, −6)

  4. For each table, determine whether the relationship is linear. If it is, find the rule. Understanding

    1. x: 0, 1, 2, 3  |  y: 1, 4, 7, 10
    2. x: 0, 1, 2, 3  |  y: 0, 1, 4, 9
    3. x: −1, 0, 1, 2  |  y: 5, 3, 1, −1
    4. x: 0, 2, 4, 6  |  y: 3, 7, 11, 15
    5. x: 1, 2, 3, 4  |  y: 3, 6, 10, 15
    6. x: 0, 1, 2, 3  |  y: −2, 1, 4, 7

  5. Describe each line without drawing it. Comment on its steepness, direction (positive/negative/zero gradient), and where it crosses the y-axis. Understanding

    1. y = 5x + 2
    2. y = −x + 3
    3. y = ¼x − 1
    4. y = −4x
    5. y = 3
    6. y = 2x − 6

  6. Linear relationships in real contexts. Problem Solving

    1. A cyclist travels at a constant speed. After 1 hour they have gone 25 km; after 3 hours, 75 km. Write the rule (distance d in terms of time t), state the gradient and what it represents in context.
    2. A phone plan charges a $20 monthly fee plus $0.10 per text message. Write a rule for total cost C in terms of number of texts n. What is the y-intercept and gradient, and what do they represent?
    3. A straight line passes through (0, −3) and (4, 5). Find the gradient, write the equation, and calculate y when x = 10.
    4. Two linear rules are y = 3x + 1 and y = 3x − 5. Without graphing, describe how these two lines are related. Do they ever intersect? Explain.

  7. Complete the table of values for y = 2x + 1, then use the graph below to check your answers. Understanding

    x−2−1012
    y = 2x + 1?????
    x y 1 2 -1 -2 1 2 3 -1 -2 -3 y = 2x+1
    1. Complete the table above.
    2. What is the gradient of the line y = 2x + 1?
    3. What is the y-intercept of y = 2x + 1?
    4. Using the graph, read off: what is y when x = 1.5?

  8. The graph below shows a straight line. Read information from the graph to answer the questions. Understanding

    x y 1 2 -1 -2 1 2 3 -1 -2
    1. Read the y-intercept from the graph. (What is y when x = 0?)
    2. Read the x-intercept from the graph. (What is x when y = 0?)
    3. Use two points on the graph to calculate the gradient.
    4. Write the equation of the line in y = mx + b form.

  9. A swimming pool is being drained. The depth (in cm) at various times is recorded below. Problem Solving

    Time (hours)01234
    Depth (cm)200175150125?
    1. Is this relationship linear? How can you tell from the table?
    2. What is the depth at t = 4 hours?
    3. Find the gradient and explain what it means in context.
    4. Write the equation for depth (D) in terms of time (t).
    5. After how many hours will the pool be completely empty (D = 0)?

  10. The graph below shows two straight lines. Study both lines carefully and answer the questions. Problem Solving

    -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 x y y = x + 3 y = x − 3
    1. What do you notice about the two lines when you look at the graph?
    2. Compare the two rules y = x + 3 and y = x − 3. How are they the same, and how are they different?
    3. Use each rule to find the y-value when x = 2. Show your working.
    4. Will the two lines ever cross? Explain your answer using the rules, not just the picture.
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