Practice Maths

Modelling Relationships 2

Key Terms

Gradient
The steepness of a straight-line graph; equals rise ÷ run between any two points; the coefficient m in y = mx + c.
y-intercept
The value of y when x = 0; where the graph crosses the y-axis; the constant c in y = mx + c.
x-intercept
The value of x when y = 0; where the graph crosses the x-axis.
Non-linear
A relationship whose graph is a curve, not a straight line; the rate of change is not constant between points.

Graphing and Interpreting Relationships

  • A linear relationship produces a straight-line graph when plotted.
  • The gradient (slope) shows the rate of change — steeper = faster change.
  • The y-intercept is where the line crosses the y-axis (when x = 0).
  • Non-linear relationships produce curved graphs (e.g., y = x² is a parabola).
  • To graph a relationship: make a table of values, plot the points, join them with a line or curve.
  • Reading graphs: find y for a given x (or x for a given y) by tracing across/up from the axes.
Hot Tip
If all plotted points lie in a straight line, the relationship is LINEAR. If the points curve, it is non-linear. The gradient equals “rise over run” between any two points on a straight line.

Worked Example 1

Graph y = 2x + 1:

Table: x = 0 → 1, x = 1 → 3, x = 2 → 5, x = 3 → 7

Plot and join → straight line, y-intercept = 1, gradient = 2.

Worked Example 2

Reading a graph: a water tank drains at 50 L per hour from 400 L.

Graph shows a straight line from (0, 400) to (8, 0).

At t = 3 hours, read off y = 250 L.

x y 0 1 2 3 0 2 4 6 y-intercept = 1 run = 1 rise = 2 y = 2x + 1 gradient = 2

Gradient = rise ÷ run = 2. y-intercept = 1 (where the line crosses the y-axis).

Using a Model to Make Predictions

One of the most powerful uses of a mathematical model is to make predictions about values you haven't measured yet. Once you have a rule (equation), substitute the input value to find the output.

  • Rule: H = 3t + 5 (height of a seedling in cm after t weeks)
  • Predict height at t = 10: H = 3(10) + 5 = 35 cm
  • When will the height reach 50 cm? 50 = 3t + 5 → 45 = 3t → t = 15 weeks

Predictions within the range of your data are called interpolation. Predictions beyond the range are called extrapolation — these are less reliable because the pattern might not continue.

Checking Whether a Model Fits the Data

Not all data follows a perfect rule. To check if a model is a good fit:

  • Substitute each given input value into the model and compare the predicted output to the actual output
  • If the predicted and actual values are close (or exactly equal), the model is a good fit
  • If there are large differences, the model may not be appropriate

For example, if the rule C = 4n predicts costs of $4, $8, $12 but the real costs are $4.10, $8.20, $12.30, the model is close but not exact — it's still useful for estimates.

Limitations of Mathematical Models

No model is perfect. There are always limitations:

  • Real data has variation and measurement error
  • A model based on a small amount of data may not apply broadly
  • The relationship might change over time (e.g. a plant eventually stops growing)
  • Models assume the rule stays constant, but real conditions change

It's important to be aware of these limitations when using models for decision-making.

Real-World Algebraic Relationships

Mathematical relationships appear everywhere. Here are a few examples to consolidate year 7 learning:

  • Distance-Speed-Time: D = S × T (distance = speed × time)
  • Area of a rectangle: A = l × w
  • Perimeter: P = 2(l + w)
  • Mobile plan costs: C = 0.15 × minutes + 20 (monthly fee plus per-minute charge)

In each case, once you know the rule, you can find any unknown by substituting the known values and solving the equation.

Key tip: When making predictions using a model, always check whether your answer makes sense in context. If a model predicts that a plant is −2 cm tall or that you need −3 litres of paint, something has gone wrong — either the model doesn't apply in that range, or there's a calculation error. Always ask "does this answer make sense?"

Mastery Practice

  1. Plot the Graph Fluency

    For each rule, complete a table of values for x = 0, 1, 2, 3, 4, then describe the graph (state whether it is a straight line, its gradient, and its y-intercept). Use the plotted graph to check.

    0 2 4 0 4 8 2 x y
    (a) y = x + 2
    0 2 4 0 4 8 12 0 x y
    (b) y = 3x
    0 2 4 0 4 8 −1 x y
    (c) y = 2x − 1
    0 2 4 0 4 4 x y
    (d) y = −x + 4
    1. y = x + 2
    2. y = 3x
    3. y = 2x − 1
    4. y = −x + 4

  2. Identify Linear or Non-Linear Fluency

    Decide if each relationship is linear or non-linear using the table of values (check whether the differences in y are constant).

    1. x: 0, 1, 2, 3 → y: 0, 1, 4, 9
    2. x: 0, 1, 2, 3 → y: 1, 3, 5, 7
    3. x: 0, 1, 2, 3 → y: 0, 2, 4, 6
    4. x: 1, 2, 3, 4 → y: 1, 4, 9, 16
    5. x: 0, 1, 2, 3 → y: 5, 5, 5, 5
    6. x: 0, 1, 2, 3 → y: 0, 1, 8, 27

  3. Reading Values from Rules Understanding

    For each rule, find the requested value.

    1. y = 4x + 3: find y when x = 5
    2. y = −2x + 10: find y when x = 3
    3. y = 3x − 5: find x when y = 7
    4. y = 5x + 1: find x when y = 26
    5. y = −x + 8: find x when y = 2
    6. y = 2x + 6: find x when y = 0

  4. Gradient and y-intercept Understanding

    For each rule y = mx + c, state the gradient (m) and y-intercept (c).

    1. y = 3x + 4
    2. y = −2x + 7
    3. y = x − 5
    4. y = ½x + 3
    5. y = −x
    6. y = 6
    7. Which of the above rules has the steepest gradient?
    8. Which of the above rules have a negative gradient?

  5. Real-World Graphs Problem Solving

    1. A runner’s distance is given by d = 8t (d in km, t in hours). What is the gradient? What does it represent? How far after 2.5 hours?
      1 2 3 4 t (h) d (km) d = 8t
    2. A phone plan costs $30 per month plus $0.20 per text. Rule: C = 0.2n + 30 (C = cost, n = texts). Find the cost for 50 texts. How many texts lead to a $50 bill?
      30 n C ($) C = 0.2n + 30
    3. Temperature decreases at 3°C per hour from 20°C: T = −3t + 20. Find T when t = 4. When does T reach 0°C?
      20°C T=0 t (h) T (°C)
    4. Two plans: Plan A costs y = 5x + 10; Plan B costs y = 3x + 20. For what value of x do they cost the same? For x = 8, which plan is cheaper?
      cross A: 5x+10 B: 3x+20 x y ($)
    5. A ball’s height (m) above the ground is given by h = −5t² + 20t (non-linear). Make a table for t = 0, 1, 2, 3, 4. Is this linear or non-linear? What is the maximum height?
      max (2, 20) t (s) h (m) parabola

  6. More Substitution into Rules Fluency

    For each rule, find the value of y when x = 0, 2, and 4.

    1. y = 6x − 3
    2. y = −4x + 16
    3. y = 2.5x
    4. y = −3x + 12
    5. y = x + 10
    6. y = 3x + 3

  7. More Find the Rule Fluency

    Find the rule y = mx + c for each table. State the gradient and y-intercept.

    1. x: 0, 1, 2, 3 → y: 2, 6, 10, 14
    2. x: 0, 1, 2, 3 → y: 9, 7, 5, 3
    3. x: 0, 1, 2, 3 → y: 0, 3, 6, 9
    4. x: 0, 1, 2, 3 → y: −1, 2, 5, 8
    5. x: 0, 2, 4, 6 → y: 5, 9, 13, 17
    6. x: 0, 5, 10, 15 → y: 100, 90, 80, 70

  8. Comparing Two Relationships Understanding

    1. Graph A has rule y = 2x + 1 and Graph B has rule y = 2x + 5. Do the lines have the same gradient? Are they parallel? Explain.
    2. Graph C has rule y = 3x + 2 and Graph D has rule y = −3x + 2. At x = 0, which graph is higher? Do the lines cross? If so, at what point?
    3. Two rules: y = 4x + 1 and y = x + 13. Find the value of x where they produce the same y-value. (Set the two rules equal and solve.)
    4. A straight-line graph has gradient 5 and passes through the point (0, −3). Write its rule. What is the y-value when x = 4?

  9. Graphing — Tables and Descriptions Understanding

    For each rule, complete a table of values for x = 0, 1, 2, 3, 4 and describe the graph fully (straight line or curve, gradient, y-intercept).

    1. y = −2x + 8
    2. y = 4x − 4
    3. y = ½x + 2
    4. y = x² (note: this is non-linear)

  10. Interpreting Context Graphs Problem Solving

    1. A water tank starts with 600 litres and drains at 75 litres per hour. Write the rule for volume V after t hours. After how many hours is the tank half-full? When is it empty?
    2. Two friends are saving money. Friend A starts with $0 and saves $25 per week. Friend B starts with $100 and saves $10 per week. Write a rule for each. After how many weeks do they have the same amount? Who has more after 10 weeks?
    3. The temperature in a freezer is given by T = −4t + 20 (T in degrees Celsius, t in hours after switching on). Find the initial temperature. Find T after 3 hours. When does the freezer reach −20°C?
    4. A bus travels along a straight road. At time t = 0, it is 80 km from town. It travels toward town at 60 km/h. Write a rule for its distance D from town. When does it arrive (D = 0)? How far is it from town at t = 0.5 h?
    5. A researcher records the relationship between the number of hours of sunshine per day x and the number of ice creams sold y at a beach kiosk. The data fits the rule y = 15x − 20. How many ice creams are predicted for 6 hours of sunshine? For how many hours does the model predict zero sales? Comment on whether negative values of y make sense in this context.
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