Cubic and Reciprocal Graphs
Cubic and reciprocal graphs are the two distinctive curve families you need to recognise and draw for Edexcel IGCSE Maths. A cubic graph comes from an equation whose highest power is x cubed, producing a sweeping S-shaped curve, while a reciprocal graph comes from dividing a constant by x, producing a two-part curve called a hyperbola that races towards the axes without ever touching them. This page shows you how to build a table of values for each, what the finished shapes look like, and how to read off key features such as the y-intercept and the asymptotes. Work through the examples, then test yourself on the auto-marked practice questions below.
Drawing a cubic graph
A cubic function has \(x^3\) as its highest power and is written \(y = ax^3 + bx^2 + cx + d\). The constant \(d\) is the \(y\)-intercept (the value of \(y\) when \(x=0\)). To draw the curve you build a table of values, plot the points, then join them with a single smooth curve.
- Choose the \(x\)-values given in the question (often a small whole-number range).
- Work out \(y\) for each \(x\) by substituting carefully. Take real care with negatives: a negative number cubed stays negative.
- Plot the points and join them with a smooth curve, not straight line segments.
Worked examples: cubic graphs
💡 Example 1: one term
\(y = x^3\) for \(-2 \le x \le 2\). Complete the table and draw the graph.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(y=x^3\) | \(-8\) | \(-1\) | \(0\) | \(1\) | \(8\) |
What is happening?
Just cube each \(x\). A negative cubed stays negative, so \((-2)^3 = -8\). This is the basic cubic shape.
💡 Example 2: add a constant
\(y = x^3 - 2\). Complete the table and draw the graph.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(x^3\) | \(-8\) | \(-1\) | \(0\) | \(1\) | \(8\) |
| \(y\) | \(-10\) | \(-3\) | \(-2\) | \(-1\) | \(6\) |
What is happening?
Cube \(x\), then subtract \(2\). The constant \(-2\) shifts the whole curve down by \(2\); the \(y\)-intercept is now \(-2\).
💡 Example 3: add a linear term
\(y = x^3 - 4x\). Complete the table and draw the graph.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(x^3\) | \(-8\) | \(-1\) | \(0\) | \(1\) | \(8\) |
| \(-4x\) | \(8\) | \(4\) | \(0\) | \(-4\) | \(-8\) |
| \(y\) | \(0\) | \(3\) | \(0\) | \(-3\) | \(0\) |
What is happening?
Work out each row, then add them for \(y\). The \(-4x\) term gives the curve its two turning points, a local maximum and a local minimum.
💡 Example 4: the full cubic
\(y = x^3 - 4x + 2\). Complete the table and draw the graph.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(x^3\) | \(-8\) | \(-1\) | \(0\) | \(1\) | \(8\) |
| \(-4x\) | \(8\) | \(4\) | \(0\) | \(-4\) | \(-8\) |
| \(+2\) | \(2\) | \(2\) | \(2\) | \(2\) | \(2\) |
| \(y\) | \(2\) | \(5\) | \(2\) | \(-1\) | \(2\) |
What is happening?
This is Example 3 with \(+2\) added to every value, so the whole curve lifts up by \(2\). The \(y\)-intercept is the constant term \(2\).
Drawing a reciprocal graph
A reciprocal function is written \(y = \dfrac{a}{x}\). The curve it produces is a hyperbola made of two separate branches. As \(x\) gets very large the curve flattens towards the \(x\)-axis, and as \(x\) gets close to \(0\) the curve shoots away up or down. The curve approaches both axes but never touches them: these axes are called asymptotes.
You cannot divide by \(0\), so there is no point on the curve at \(x=0\). Choose \(x\)-values on both sides of zero so you capture both branches.
Shifted reciprocals. Some reciprocal graphs have the curve moved off the axes. In \(y = \dfrac{a}{x} + d\) every \(y\)-value is \(d\) bigger, so the whole curve slides up (or down when \(d\) is negative) and the horizontal asymptote moves from \(y = 0\) to \(y = d\). In \(y = \dfrac{a}{x - h}\) the curve slides sideways: you cannot divide by \(0\), so now it is \(x = h\) that is undefined and the vertical asymptote moves there. Work these out exactly as before, one \(x\)-value at a time.
Worked example: reciprocal graph
💡 Example 5: reciprocal
\(y = \dfrac{12}{x}\). Complete the table and draw the graph.
| \(x\) | \(-6\) | \(-3\) | \(-2\) | \(2\) | \(3\) | \(6\) |
|---|---|---|---|---|---|---|
| \(y\) | \(-2\) | \(-4\) | \(-6\) | \(6\) | \(4\) | \(2\) |
What is happening?
Divide \(12\) by each \(x\). Positive \(x\) gives the top-right branch, negative \(x\) the bottom-left branch. There is no value at \(x = 0\).
🔑 Key points
- Cubic: highest power \(x^3\), form \(y = ax^3 + bx^2 + cx + d\), an S-shaped curve.
- If \(a>0\) the cubic rises left to right; if \(a<0\) it falls.
- The \(y\)-intercept of a cubic is the constant term \(d\).
- Reciprocal: form \(y = \dfrac{a}{x}\), a hyperbola of two branches.
- The axes are asymptotes: the reciprocal curve gets ever closer but never touches them.
- Shifting moves the asymptotes with the curve: \(y = \dfrac{a}{x} + d\) has asymptote \(y = d\), and \(y = \dfrac{a}{x - h}\) is undefined at \(x = h\).
⚠️ Common pitfalls
- Joining cubic points with straight segments instead of one smooth curve.
- Sign slips: \((-2)^3 = -8\), not \(8\). Cube the whole negative.
- Trying to plot \(x=0\) on a reciprocal graph; it is undefined there.
- Assuming \(x = 0\) is always the undefined value: for \(y = \dfrac{a}{x - h}\) it is \(x = h\).
- Drawing the two reciprocal branches as if they connect; they are separate.
- Letting a reciprocal branch touch or cross an axis.