Practice Maths

Topic Review — Linear and Non-Linear Relationships

Mixed Practice — L15 – L17

This review covers all lessons in the Linear and Non-Linear Relationships topic. Try each question before checking your answers.

Review Questions

  1. Graphing linear relationships. Fluency

    1. State the gradient and y-intercept: y = −4x + 9
    2. State the gradient and y-intercept: y = ⅔x − 5
    3. Complete a table of values for y = 3x − 2 using x = −2, −1, 0, 1, 2.
    4. Find the x-intercept and y-intercept of 2x + 5y = 20.
    5. Write the equation of a line with gradient −3 and y-intercept 7.
    6. Are y = 5x − 1 and y = 5x + 8 parallel? Explain.

  2. Non-linear relationships. Fluency

    1. Complete the table for y = x2 − 3 using x = −2, −1, 0, 1, 2.
    2. Complete the table for y = 1/x using x = −4, −2, −1, 1, 2, 4.
    3. For y = x2 + 1, state: (a) the vertex, (b) the axis of symmetry, (c) the y-intercept.
    4. Evaluate y = 2x for x = −1, 0, 1, 2, 3, 4.

  3. Classify and compare. Understanding

    1. Classify each as linear or non-linear and give a reason:
      (a) y = −7x + 2    (b) y = x3 + x    (c) y = 4    (d) y = 5/x    (e) y = 10x
    2. Use first differences to determine whether the relationship in this table is linear or non-linear. If linear, write the equation.
      x01234
      y62−2−6−10
    3. Use first differences to classify:
      x01234
      y36122448
    4. Describe in words how the graphs of y = x and y = x2 differ.

  4. Finding equations and sketching. Understanding

    1. Find the equation of the line through (0, 5) and (3, 14).
    2. Find the equation of the line through (−2, 6) and (4, −6).
    3. Sketch y = −2x + 4 on a clearly labelled set of axes, showing the x-intercept, y-intercept, and one additional point.
    4. Sketch y = x2 − 1 on clearly labelled axes, labelling the vertex and x-intercepts.
    5. A line is parallel to y = 3x − 2 and passes through the point (0, 5). Write its equation.

  5. Problem solving. Problem Solving

    1. A pool holds 24 000 litres. It is drained at a constant rate of 800 litres per hour.
      1. Write an equation for the volume V (litres) remaining after t hours.
      2. How long does it take to fully drain the pool?
      3. Sketch the graph and describe its key features.
    2. The profit P (dollars) from selling x items is modelled by two different plans:
      Plan X: P = 8x − 40
      Plan Y: P = x2 − 2x
      1. Which plan is linear and which is non-linear?
      2. Calculate the profit for both plans when x = 10.
      3. At what value of x do both plans give the same profit? (Hint: set equal and solve.)
    3. A student claims that y = x2 and y = x are the same when x = 0 and x = 1. Are they correct? What happens for x = 2, 3, 4? What does this illustrate about comparing linear and non-linear relationships?
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