Practice Maths

Non-Linear Relationships

Key Ideas

Key Terms

non-linear relationship
Has a curved graph — the rate of change is not constant.
Quadratic
Y = x2: a parabola; symmetric about the y-axis; vertex at (0, 0); minimum point.
Hyperbola
Y = k/x: two branches; asymptotes at x = 0 and y = 0; never touches either axis.
Exponential
Y = ax: always positive; passes through (0, 1); increases rapidly for positive x; approaches y = 0 for negative x (horizontal asymptote).
axis of symmetry
Of y = x2 is x = 0 (the y-axis).
EquationGraph typeKey feature
y = x2ParabolaVertex (0, 0); symmetric about y-axis
y = x2 + cParabolaVertex (0, c); shifted up/down
y = (x − h)2ParabolaVertex (h, 0); shifted left/right
y = k/xHyperbolaAsymptotes x = 0 and y = 0
y = axExponentialPasses through (0, 1); asymptote y = 0
Hot Tip For y = x2, symmetry means (−3)2 = 32 = 9. Always substitute negative x-values carefully — the square of a negative is positive.

Worked Example

Question: Complete a table of values for y = x2 for x = −3 to 3, then state the vertex and axis of symmetry.

x−3−2−10123
y = x29410149

Vertex: (0, 0) — the minimum point.   Axis of symmetry: x = 0.

What Makes a Relationship Non-Linear?

A linear relationship produces a straight line graph and has a constant rate of change — every time x increases by 1, y increases by the same fixed amount. A non-linear relationship does not have a constant rate of change, so its graph is curved. The three main non-linear types you need to know in Year 9 are parabolas, hyperbolas, and exponentials.

Parabolas: y = x2

The equation y = x2 produces a U-shaped curve called a parabola. Key features: it is symmetric about the y-axis, has a minimum point (the vertex) at (0, 0), and is never negative (y ≥ 0 for all x). As x gets large (positive or negative), y grows very quickly. Real-world example: the path of a ball thrown through the air follows a parabolic shape. Variations like y = x2 + 3 shift the parabola up by 3 units.

Hyperbolas: y = 1/x

The equation y = 1/x produces a hyperbola — a curve with two separate branches, one in the first quadrant (both x and y positive) and one in the third quadrant (both negative). As x gets large, y gets close to zero but never reaches it (the x-axis is an asymptote). As x approaches zero, y gets extremely large. Note: x = 0 is undefined, so there is a gap in the graph at x = 0. Real-world example: the time taken to travel a fixed distance decreases as speed increases — time = distance/speed.

Exponentials: y = 2x

The equation y = 2x produces an exponential curve. Key features: it is always positive (y > 0 for all x), passes through (0, 1) since 20 = 1, and rises steeply for large x. For negative x, y gets close to zero but never reaches it (the x-axis is an asymptote again). Exponential relationships model rapid growth: population growth, compound interest, and the spread of a virus all follow exponential patterns.

Key tip: A quick way to tell the three curves apart is to look at what happens as x gets large. Parabolas grow fast but come back to zero at x = 0. Hyperbolas approach zero from above. Exponentials grow faster and faster without limit. On the negative side, parabolas are still positive, hyperbolas become large and negative, and exponentials approach zero.

Building Tables of Values

To graph any non-linear relationship, build a table of values by substituting x-values and calculating y. For y = x2, use x = −3, −2, −1, 0, 1, 2, 3 to get y = 9, 4, 1, 0, 1, 4, 9. Plot each point carefully, then join them with a smooth curve — never connect the dots with straight segments. Including negative x-values and zero is essential to show the full shape of the curve.

Mastery Practice

  1. Complete the table of values for y = x2 using x = −3, −2, −1, 0, 1, 2, 3. State the vertex and axis of symmetry. Fluency

    x−3−2−10123
    y = x2       

  2. Complete the tables of values for the two equations below (x = −3 to 3), then describe how each differs from y = x2. State the vertex of each. Fluency

    (a) y = x2 + 2

    x−3−2−10123
    y = x2 + 2       

    (b) y = (x − 1)2

    x−3−2−10123
    y = (x − 1)2       

  3. Identify the graph type for each equation. Choose from: linear, parabola, hyperbola, exponential. Fluency

     EquationGraph type
    (a)y = 3x + 1 
    (b)y = x2 − 4 
    (c)y = 6/x 
    (d)y = 5x 
    (e)y = (x + 2)2 
    (f)y = −4/x 
    (g)y = 2x 
    (h)y = −x2 + 5 

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

    1. The graph of y = x2 is symmetric about the y-axis.
    2. The hyperbola y = 1/x passes through the origin.
    3. The exponential y = 3x has a y-intercept of (0, 1).
    4. The vertex of y = x2 + 5 is (0, 5).
    5. The graph of y = (x − 3)2 has its vertex at (3, 0).
    6. A hyperbola y = k/x has a horizontal asymptote at x = 0.

  5. Describe each transformation of y = x2. State the vertex, direction of opening, and whether the parabola is narrower or wider than y = x2. Understanding

    Transformation Guide. Adding a constant c shifts up/down; replacing x with (x − h) shifts left/right; multiplying by a > 1 narrows; a negative coefficient flips the parabola.
    1. y = x2 + 4
    2. y = x2 − 3
    3. y = (x − 2)2
    4. y = −x2
    5. y = 3x2
    6. y = −x2 + 6

  6. Find the y-value for the given x-value in each equation. Show your working. Understanding

    Substitution. Replace x with the given value and evaluate carefully, especially with negatives inside squares or exponentials.
    1. y = x2 − 5 when x = 4
    2. y = (x − 3)2 when x = 1
    3. y = 8/x when x = −4
    4. y = 2x when x = 5
    5. y = −x2 + 10 when x = −3
    6. y = (x + 1)2 − 4 when x = 2

  7. A graph shows the height h (metres) of a jumping ball over time t (seconds), modelled by h = −t2 + 6t. Understanding

    Ball Trajectory. The height of the ball follows h = −t2 + 6t. The graph is an inverted parabola.
    1. Complete a table of values for t = 0, 1, 2, 3, 4, 5, 6.
    2. What is the maximum height of the ball? At what time does this occur?
    3. At what times does the ball touch the ground (h = 0)?
    4. What is the axis of symmetry of this parabola?
    5. Is the parabola opening up or down? How can you tell from the equation?

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

    Comparing graphs. Think about shape, intercepts, symmetry, and whether the graph touches the axes or has asymptotes.
    1. y = x2 versus y = 2x
    2. y = x2 + 1 versus y = −x2 + 1
    3. y = 4/x versus y = 4x

  9. Projectile motion. Problem Solving

    Thrown Ball. A ball is thrown and its height (metres) is modelled by h = −x2 + 4x, where x is horizontal distance in metres.
    1. Complete a table of values for x = 0, 1, 2, 3, 4.
    2. What is the maximum height of the ball, and at what horizontal distance does it occur?
    3. At what horizontal distances is the ball on the ground (h = 0)? Solve h = −x2 + 4x = 0 by factorising.
    4. State the axis of symmetry and the vertex of the parabola.
    5. If the ball must clear a fence of height 3 m, for what range of x-values is the ball above the fence?

  10. Comparing non-linear relationships. Problem Solving

    Bacteria Growth vs. Square Area. Two quantities are modelled: bacteria count N = 2t (t in hours), and area of a square A = s2 (s in cm).
    1. For N = 2t, complete a table for t = 0, 1, 2, 3, 4, 5.
    2. For A = s2, complete a table for s = 0, 1, 2, 3, 4, 5.
    3. Both models are non-linear. Explain one key difference between the shapes of their graphs.
    4. At t = 5 (or s = 5), which quantity is growing faster? Justify using your tables.
    5. A colony starts with N = 20 = 1 bacterium. How many hours until there are more than 100 bacteria? Show your reasoning.
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