Cubic and Reciprocal Graphs

Beyond straight lines and parabolas, two more standard graph families appear at GCSE: cubic functions (highest power 3) and the reciprocal function \( y = \frac{1}{x} \). Both have distinctive shapes that can be recognised, sketched and interpreted.

Cubic Functions

The general form is \( y = ax^3 + ldots \). The key features of a cubic graph are:

For \( a > 0 \): rises from bottom-left to top-right, with a characteristic S-shaped inflection at the origin for \( y = x^3 \).
For \( a < 0 \): falls from top-left to bottom-right (reflected).
A cubic with three distinct linear factors (e.g. \( y = x(x-2)(x+2) \)) crosses the x-axis at each root with a direction change.
A repeated root gives a point where the graph touches but does not cross the x-axis.

The Reciprocal Function \( y = frac{1}{x} \)

The reciprocal function is defined for all \( x eq 0 \) and has two key features:

Two separate branches: one in Quadrant I (\( x>0, y>0 \)) and one in Quadrant III (\( x<0, y<0 \)).
Asymptotes: the x-axis (\( y = 0 \)) and y-axis (\( x = 0 \)). The graph approaches but never touches either axis.

As \( x \to 0^+ \), \( y \to +\infty \); as \( x \to +\infty \), \( y \to 0^+ \). The shape is symmetric about the lines \( y = x \) and \( y = -x \).

Worked Examples

Sketch \( y = x(x-3)(x+2) \), marking roots and general shape.

Roots: \( x = 0, x = 3, x = -2 \). Leading term: \( x cdot x cdot x = x^3 \), so \( a = 1 > 0 \) — rises from bottom-left to top-right.

Sketch: starts from bottom-left, crosses x-axis at \(-2\), rises to a local max, falls to cross at \(0\), continues falling to a local min, then crosses again at \(3\) and rises to top-right.

y-intercept: \( x=0 \Rightarrow y=0 \).

Describe the graph of \( y = \dfrac{1}{x} \) and explain why \( x = 0 \) is excluded.

Two separate branches in Quadrants I and III. Asymptotes at \( x=0 \) and \( y=0 \): the graph approaches both axes but never reaches them.

\( x=0 \) is excluded because division by zero is undefined — the function has no value there. The y-axis is therefore a vertical asymptote.

How does \( y = -x^3 \) differ from \( y = x^3 \)? How does \( y = \dfrac{-1}{x} \) differ from \( y = \dfrac{1}{x} \)?

\( y = -x^3 \): reflected in the x-axis. The S-curve now falls from top-left to bottom-right. Roots, asymptotes: same structure but mirrored.

\( y = -\frac{1}{x} \): branches in Quadrants II and IV instead of I and III. Asymptotes remain at \( x=0 \) and \( y=0 \). The graph is reflected in the x-axis (or equivalently in the y-axis).

Key Takeaways

Cubic \( y = ax^3 + ldots \): for \( a > 0 \) the graph rises left-to-right in an S-shape; for \( a < 0 \) it falls left-to-right.
A cubic can have 1, 2 or 3 x-intercepts; it always has exactly one y-intercept.
Reciprocal \( y = 1/x \): two branches in Q1 and Q3; asymptotes along both axes; never crosses either axis.
Multiplying by \(-1\) reflects the graph in the x-axis: \( y=-x^3 \) falls left-to-right; \( y=-1/x \) has branches in Q2 and Q4.