O-Level E Maths tests 6 graph families: y = ax (line), y = ax² (parabola), y = ax³ (cubic), y = a/x (hyperbola), y = a/x², and y = kaˣ (exponential). For each one, the sign of the coefficient (a or k) flips the graph's orientation — the shape stays the same, only its direction changes. Recognising the family from the equation, then applying the correct sign rule, is the entire skill.
Sketching these graphs doesn't require plotting multiple points and joining them — that's slow and error-prone under exam conditions. Instead, O-Level expects you to recognise which of the 6 families an equation belongs to, then sketch the correct general shape from memory, adjusted for the sign of the coefficient. Once you know the 6 shapes and their sign rules, you can sketch any of them in under 30 seconds.
a > 0
a < 0
A straight line through the origin (0,0). When a > 0, the line rises from bottom-left to top-right. When a < 0, it falls from top-left to bottom-right. Larger |a| means a steeper line.
a > 0 (U-shaped, minimum at origin)
a < 0 (∩-shaped, maximum at origin)
A parabola with its turning point at the origin. When a > 0, it's U-shaped (opens upward, minimum at origin). When a < 0, it's ∩-shaped (opens downward, maximum at origin). Symmetric about the y-axis.
a > 0
a < 0
An S-shaped curve passing through the origin, flattening briefly at (0,0). When a > 0, it rises overall from bottom-left to top-right. When a < 0, it falls overall from top-left to bottom-right — a mirror image of the a > 0 case, reflected in the x-axis.
a > 0 (quadrants 1 & 3)
a < 0 (quadrants 2 & 4)
Two separate curved branches that never touch the axes (asymptotic). When a > 0, the branches sit in the top-right and bottom-left (quadrants 1 and 3). When a < 0, they sit in the top-left and bottom-right (quadrants 2 and 4).
a > 0 (both branches above x-axis)
a < 0 (both branches below x-axis)
Similar to y = a/x, but since x² is always positive, both branches sit on the same side of the x-axis — both above when a > 0, both below when a < 0. This is the key visual difference from y = a/x, where the two branches sit diagonally opposite.
k > 0 (growth, y-intercept at k)
k < 0 (reflected in x-axis)
A curve that never touches the x-axis (asymptotic to y = 0), always crossing the y-axis at (0, k). When k > 0, the curve rises steeply to the right (exponential growth) while approaching but never reaching y = 0 on the left. When k < 0, the whole curve is reflected below the x-axis.
Sketch the graph of y = −3/x².
Solution
The fastest way to identify the family: look at where x appears in the equation. x alone → line. x² → parabola or /x² hyperbola-type (check if x² is in the numerator or denominator). x³ → cubic. x in the exponent (like 2ˣ) → exponential. This single check narrows down the family before you even look at the sign.
Do I need to plot points to sketch these graphs in the exam?
Not usually — O-Level questions typically ask for a sketch showing the correct general shape, key features (intercepts, asymptotes, turning points), and correct orientation based on the sign. A small number of key points (like the y-intercept) may be expected, but a full table of values and precise plotting usually isn't required unless the question specifically says "draw an accurate graph." When a question asks you to read off a solution from the sketch, that's the skill covered in solving equations graphically.
What's the difference between y = a/x and y = a/x²?
For y = a/x, the sign of y depends on the sign of x — so the two branches sit in diagonally opposite quadrants. For y = a/x², x is squared, making the denominator always positive regardless of x's sign — so both branches sit on the same side of the x-axis, determined only by the sign of a.
Why does y = kaˣ never touch the x-axis?
Because aˣ (with a > 1) is always positive for any real x — it can get extremely close to 0 as x becomes very negative, but never actually reaches 0. This means y = kaˣ approaches the x-axis but never touches it, making y = 0 a horizontal asymptote.
How do I remember which shape is which under exam pressure?
Link each family to something familiar: y = ax is a straight line (simplest case), y = ax² is the parabola from completing the square, y = ax³ is a stretched "S", y = a/x and y = a/x² are both "broken" curves that avoid the axes, and y = kaˣ is exponential growth or decay — the only family with x as a power rather than a base.
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Mr. Gan drills the family-first method until you can sketch any of the 6 shapes on sight, so graph questions stop eating your exam time.
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