Practice Maths

Comparing Linear and Non-Linear Relationships

Key Ideas

Key Terms

linear
If its graph is a straight line and the gradient is constant.
first difference test
Subtract consecutive y-values. If all differences are equal, the relationship is linear.
non-linear
If the first differences are not equal — the graph will be curved.
quadratic
The second differences are constant (first differences form an arithmetic sequence).
power of 1
.
FeatureLinearNon-Linear
Graph shapeStraight lineCurved
GradientConstantChanges
First differencesEqualNot equal
Power of xExactly 1≠ 1, or x in denominator/exponent
Hot Tip The first difference test: list y-values in order and subtract consecutive pairs. Equal differences ⇒ linear. Unequal differences ⇒ non-linear.

Worked Example

Table A (linear):

x01234
y357911

First differences: 5−3=2, 7−5=2, 9−7=2, 11−9=2. All equal ⇒ Linear. Gradient m=2, y-intercept b=3, so y = 2x + 3.

Identifying Type from a Table of Values

Given a table of x and y values, check whether the relationship is linear by calculating the first differences — the change in y each time x increases by 1. If the first differences are all equal (constant), the relationship is linear. If the first differences keep changing, the relationship is non-linear. For example, if y values are 3, 5, 7, 9 (differences all +2), that is linear. If y values are 1, 4, 9, 16 (differences +3, +5, +7), that is non-linear.

Note: the first-difference test only works reliably when x-values are evenly spaced (each step is the same size). If the x-values jump unevenly, you need to check the ratio of change in y to change in x each time.

Identifying Type from a Graph

On a graph, the distinction is straightforward: a straight line is always linear, and any curve is non-linear. When examining a graph, look for bends, turns, or branches. A parabola has a single turning point; a hyperbola has two branches that approach the axes; an exponential curve gets steeper as it moves in one direction and flattens toward zero in the other. If the graph is completely straight, with no bends, it is linear.

Identifying Type from an Equation

You can also classify by looking at the equation. If x appears only to the power of 1 (no x2, no 1/x, no 2x), the relationship is linear. Examples of linear equations: y = 3x + 1, y = −x + 4, 2x + 3y = 6. Examples of non-linear equations: y = x2 + 2 (parabola), y = 1/x (hyperbola), y = 3x (exponential). Even if the equation looks unusual, just ask: does x appear as a power greater than 1, as a denominator, or as an exponent?

Key tip: When asked to justify your classification, always give the specific reason — do not just say "it looks curved." Write something like: "The first differences are not constant (they are +3, +5, +7), so the relationship is non-linear" or "The equation contains x2, which means the graph is a parabola, not a straight line."

Mixed Classification Problems

Exam questions often present a mixture of graphs, tables, and equations in one question and ask you to classify each as linear or non-linear. Work through each representation using the method that suits it: first differences for tables, shape for graphs, and power of x for equations. If you are given a real-world context — for example, "the area of a square with side length x" — recognise that A = x2 is non-linear even before drawing a graph.

Mastery Practice

  1. Classify each equation as linear or non-linear, and give the reason. Fluency

     EquationLinear / Non-LinearReason
    (a)y = 5x − 3  
    (b)y = x2 + 1  
    (c)y = 3/x  
    (d)y = −4x + 7  
    (e)y = 2x  
    (f)y = x3 − 2  
    (g)y = 9  
    (h)y = ½x + 9  

  2. Calculate the first differences for each table and state whether the relationship is linear or non-linear. Fluency

     xyFirst differencesLinear / Non-Linear
    (a)0, 1, 2, 3, 42, 6, 10, 14, 18  
    (b)0, 1, 2, 3, 40, 1, 4, 9, 16  
    (c)1, 2, 3, 4, 5−3, 1, 5, 9, 13  
    (d)0, 1, 2, 3, 41, 3, 9, 27, 81  
    (e)−2, −1, 0, 1, 25, 3, 1, −1, −3  

  3. Match each equation to the correct graph description below. Write the letter next to each description. Fluency

    Equations:   A: y = 3x − 1    B: y = x2 − 2    C: y = 4/x    D: y = 2x

     DescriptionEquation (A/B/C/D)
    (i)Straight line with positive gradient and negative y-intercept 
    (ii)U-shaped parabola with vertex at (0, −2) 
    (iii)Two-branch curve; asymptotes on both axes; defined only for x ≠ 0 
    (iv)Always positive; passes through (0, 1); rises rapidly for large positive x 

  4. State whether each statement is True or False. If false, write a corrected version. Fluency

    1. A linear graph always passes through the origin.
    2. If the first differences of a table of values are all equal, the relationship is linear.
    3. y = −3x + 5 is non-linear because it has a negative gradient.
    4. The graph of y = x2 is a straight line.
    5. An exponential relationship y = 2x has a constant rate of change.
    6. For a quadratic relationship, the second differences are constant.

  5. For each real-world situation, decide whether the relationship is linear or non-linear. Justify your answer. Understanding

    Real-World Contexts. Think about whether the rate of change is constant or varies as the input grows.
    1. A person earns $18 per hour at a casual job. Is their total earnings versus hours worked linear or non-linear?
    2. A ball is dropped from a height. Its speed (m/s) increases as v = 10t, where t is time in seconds. Is this linear or non-linear?
    3. A company triples its revenue each year. Is revenue versus year linear or non-linear?
    4. Water drips into a bucket at 2 mL per second. Is volume versus time linear or non-linear?

  6. For each table, calculate first differences, classify as linear or non-linear, and — if linear — write the equation in y = mx + b form. Understanding

    Finding the Equation. If linear: the common first difference is the gradient m. The y-intercept b is the y-value when x = 0.
    1. x0123
      y471013
    2. x0123
      y02818
    3. x0123
      y−1258
    4. x0123
      y24816

  7. For each pair of equations, explain two key differences you would expect to see between their graphs. Understanding

    Graph Comparison. Think about shape, whether the graph curves or is straight, intercepts, and rate of change.
    1. y = 2x + 1 versus y = 2x2 + 1
    2. y = x + 3 versus y = 3x
    3. y = −x + 4 versus y = −x2 + 4

  8. Classify the relationship based on the given information. Suggest a possible equation. Understanding

    Pattern Recognition. Use what you know about first differences and graph shapes to identify the type of relationship.
    1. The graph is a straight line passing through (0, −2) with a positive slope.
    2. The graph is a symmetric U-shape with the lowest point at (0, 1).
    3. The first differences in a table are: −3, −3, −3, −3.
    4. The first differences in a table are: 2, 4, 6, 8 (increasing by 2 each time).

  9. Comparing costs. Problem Solving

    Mobile Plans. Plan A: $30 flat fee + $0.10 per minute. Plan B costs are shown in the table below.
    Minutes (m)050100150200
    Cost $ (Plan B)10204080160
    1. Write an equation for Plan A’s cost C in terms of minutes m.
    2. Is Plan A linear or non-linear? Justify using the equation.
    3. Calculate first differences for Plan B’s costs and classify it as linear or non-linear. Justify your answer.
    4. Which plan is cheaper for 100 minutes? Show your calculations.
    5. For a very high number of minutes, which plan becomes much more expensive? Explain using the nature of the relationship.

  10. Area and perimeter comparison. Problem Solving

    Square Garden. A square garden has side length s metres. Its area is A = s2 and its perimeter is P = 4s.
    1. Complete tables of values for s = 1, 2, 3, 4, 5 for both A and P.
    2. Use first differences to determine whether each relationship is linear or non-linear.
    3. By how much does the area increase when the side goes from 4 m to 5 m? Compare with the increase from 1 m to 2 m.
    4. By how much does the perimeter increase when the side goes from 4 m to 5 m? Compare with the increase from 1 m to 2 m.
    5. A student says: “All equations with a negative in them are non-linear.” Explain why this is incorrect using one linear example and one non-linear example.
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