Pearson Further Pure Mathematics 4: Graphs

A graph is a structured argument, not a decorative curve. Its intercepts, symmetry, end behavior, asymptotes and stationary points encode algebraic facts. Pearson International GCSE Further Pure Mathematics (4PM1) expects you to recognise standard graphs, transform them, locate intersections, and use sketches to solve equations or inequalities. A credible sketch need not be drawn to scale, but every labelled feature must agree with the function's domain and behavior.

1. Build a sketch from features

Before drawing, identify the graph family and make a feature list. Useful questions are:

• What values of x are allowed, and what outputs are possible?
• Where does the graph meet either axis?
• Are there asymptotes, turning points or points of inflection?
• Is the graph even, odd, periodic or otherwise symmetric?
• What happens as x becomes very large, very negative or approaches an excluded value?

A polynomial is defined for every real x and has no vertical asymptote. Its leading term controls end behavior. An even-degree polynomial has ends moving in the same direction; an odd-degree polynomial has ends moving in opposite directions. Real roots locate horizontal-axis intersections. A repeated root of even multiplicity usually gives tangency, while an odd-multiplicity root gives a crossing.

The reciprocal graph y = 1/x has domain and range excluding zero, with asymptotes x = 0 and y = 0. Its branches occupy the first and third quadrants. The graph y = 1/x² is positive and even, so its branches occupy the first and second quadrants. An asymptote describes limiting behavior. The curve may approach it indefinitely without meeting it in the standard reciprocal cases; do not join branches through an excluded input.

For a base a > 1, y = aˣ is positive, increasing, passes through (0, 1) and approaches the horizontal asymptote y = 0 as x decreases. Its inverse y = log_a x has domain x > 0, passes through (1, 0) and has vertical asymptote x = 0. The two graphs reflect in y = x. For 0 < a < 1, both monotonic directions reverse, but their domains and inverse relationship remain.

2. Translations

If y = f(x) is known, then y = f(x) + k translates it vertically by vector (0, k). Every output changes by k, so horizontal asymptotes and the range move with the graph. The graph