Practice Maths

★ Topic Review — Modelling Relationships

This review covers Lessons 68–69.

Review Questions

  1. Rules and Tables Fluency

    1. Find the rule for: x: 0, 1, 2, 3 → y: 4, 7, 10, 13
    2. Complete the table for y = 5x − 3: x = 1, 2, 3, 4, 5
    3. Find the rule for: x: 1, 2, 3, 4 → y: 7, 14, 21, 28
    4. Is y = 3x proportional? Is y = 3x + 2 proportional? Explain.

  2. Linear or Non-Linear Fluency

    State linear or non-linear for each table of values:

    1. x: 0, 1, 2, 3 → y: 3, 7, 11, 15
    2. x: 0, 1, 2, 3 → y: 0, 1, 4, 9
    3. x: 1, 2, 3, 4 → y: 2, 4, 6, 8
    4. x: 0, 1, 2, 3 → y: 1, 2, 4, 8

  3. Gradient and y-intercept Understanding

    For y = 4x − 3 and y = −2x + 8:

    1. Find the gradient of each.
    2. Find the y-intercept of each.
    3. Find y when x = 3 for each rule.
    4. Find x when y = 0 for each rule (the x-intercept).

  4. Writing Rules from Context Understanding

    1. A pool is being filled at 200 litres per minute starting empty. Write a rule for volume V (litres) after t minutes. How full after 15 minutes?
    2. A car rental costs $50 per day plus $0.25 per km. Write a rule for total cost C in terms of km travelled (k). Find the cost for 120 km.
    3. A student earns $12 per hour babysitting. Write a rule for total earnings E after h hours. How many hours to earn $90?
    4. A candle is 20 cm tall and burns at 2 cm per hour. Write a rule for height H after t hours. When will the candle burn out?

  5. Problem Solving Problem Solving

    1. Two phone plans: Plan A: C = 0.15n + 25; Plan B: C = 0.10n + 35. Which is cheaper for 100 texts? For 200 texts? At what number of texts are they equal?
    2. A table has rule y = ax + b. When x = 2, y = 11. When x = 5, y = 20. Find a and b. Then find y when x = 10.
    3. A rule is y = 3x + 5. Another rule is y = 3x − 2. Both have the same gradient. Explain what this means graphically. Which line is higher on the y-axis? When x = 4, find both y-values.
    4. A non-linear table: x: 0, 1, 2, 3 → y: 0, 1, 4, 9. Continue the table to x = 5. Is this pattern linear? Write the rule in the form y = x?.

  6. Read values from a rule: Fluency

    Use the rule C = 3n + 10 (cost in dollars for n items):

    1. Find C when n = 0.
    2. Find C when n = 8.
    3. Find n when C = 40.
    4. What does the 10 represent in the context of this rule?

  7. Direct proportion: Understanding

    1. y is directly proportional to x, and y = 21 when x = 3. Find the rule y = kx.
    2. Use your rule to find y when x = 7.
    3. A recipe uses 150 g of sugar per 500 g of flour. Write a rule for the amount of sugar S needed for F grams of flour.
    4. How much sugar is needed for 750 g of flour?

  8. Comparing linear rules: Understanding

    Plumber A charges $60 call-out fee plus $80/hour. Plumber B charges $100 call-out fee plus $60/hour.

    1. Write a cost rule for each plumber in terms of hours h.
    2. Find the cost of each plumber for a 3-hour job.
    3. For how many hours are the costs equal? Show your working.
    4. Which plumber would you choose for a 5-hour job?

  9. Interpret a graph: Understanding

    A graph of distance vs time shows a straight line passing through (0, 0) and (4, 120).

    1. What is the gradient (rate of change)?
    2. What does the gradient represent in context?
    3. Write the rule for distance d after time t hours.
    4. How far would be travelled in 6.5 hours?

  10. Graph features: Problem Solving

    1. A rule y = −2x + 8 is graphed. Find the x-intercept and y-intercept. Describe what they mean if x represents time (hours) and y represents water level (cm).
    2. Two rules are graphed: y = 2x + 1 and y = 2x − 3. Describe the relationship between the two lines. Will they ever intersect? Explain.
    3. A rule has a gradient of −3 and a y-intercept of 15. Write the rule. What is the x-intercept?
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