Graphs for Edexcel IGCSE Further Pure Mathematics: Methods and Worked Examples

Graphs in edexcel igcse further pure mathematics test your ability to visualise algebraic expressions and use curve sketching to solve equations

Graphs are the visual language of mathematics. In this section of the specification, you are required to sketch and interpret graphs of polynomials and rational functions with linear denominators, understand the concept of asymptotes parallel to the coordinate axes, and use graphical methods to solve equations. These are distinct skills that the exam tests in distinct ways: sketching asks you to produce a clear diagram with labelled features, interpreting asks you to read information from a given graph, and graphical solving asks you to find where two curves intersect.

These edexcel igcse further pure mathematics graphs revision notes cover both sub-topics with worked examples, key diagrams, and common mistakes. The precision required in graph sketching often surprises students who treat it as approximate artwork. It is not. It is algebra made visible, and the exam awards marks for specific features.

Graphs of polynomials

A polynomial graph has no breaks, no jumps and no asymptotes. Its degree determines its overall shape:

What the exam expects in a polynomial sketch

• x-intercepts: factorise the polynomial to find the roots, and plot them. If a root has multiplicity 2 (repeated factor), the curve touches the x-axis and turns back. If the root has multiplicity 1, the curve crosses.
• y-intercept: evaluate the polynomial at x = 0.
• End behaviour: determined by the degree and the sign of the leading coefficient.
• Turning points: if calculus is appropriate, differentiate to find stationary points. For a sketch question in the Graphs section, you may sometimes be expected to use the factorised form rather than calculus.

Worked example: sketching a cubic

Sketch y = (x + 1)(x - 2)(x - 4).

x-intercepts: x = -1, x = 2, x = 4 (all multiplicity 1, so the curve crosses at each).

y-intercept: y = (1)(-2)(-4) = 8.

Leading term: x³ (positive coefficient), so the curve goes from bottom-left to top-right.

The sketch shows a cubic that crosses the x-axis at -1, 2 and 4, passes through (0, 8), and has two turning points between the roots.

Rational functions with linear denominators

A rational function with a linear denominator has the form f(x) = (ax + b) / (cx + d) or more generally f(x) = (polynomial) / (linear factor). These functions have asymptotes, which are the defining feature of their graphs.

Vertical asymptotes

A vertical asymptote occurs where the denominator equals zero (and the numerator does not equal zero at the same point). For y = (2x + 1) / (x - 3), the vertical asymptote is x = 3.

Horizontal asymptotes

For a function of the form y = (ax + b) / (cx + d), the horizontal asymptote is y = a/c. As x becomes very large (positive or negative), the constant terms become negligible and y approaches a/c. For y = (2x + 1) / (x - 3), the horizontal asymptote is y = 2.

Worked example: sketching a rational function

Sketch y = (3x - 1) / (x + 2), showing all asymptotes and intercepts.

Vertical asymptote: x + 2 = 0, so x = -2.

Horizontal asymptote: y = 3/1 = 3.

x-intercept: set 3x - 1 = 0, so x = 1/3. The curve crosses the x-axis at (1/3, 0).

y-intercept: set x = 0, so y = -1/2. The curve crosses the y-axis at (0, -1/2).

The curve has two branches: one in the region x < -2 (approaching x = -2 from the left and y = 3 as x goes to negative infinity) and one in the region x > -2 (approaching x = -2 from the right and y = 3 as x goes to positive infinity). The curve never crosses either asymptote in this case.

Worked example: rational function with a quadratic numerator

Sketch y = (x² - 4) / (x - 1).

Simplify: x² - 4 = (x - 2)(x + 2), so y = (x - 2)(x + 2) / (x - 1). This does not simplify further (no common factors).

Vertical asymptote: x = 1.

Since the degree of the numerator (2) is one more than the degree of the denominator (1), there is an oblique (slant) asymptote rather than a horizontal one. Perform polynomial division: x² - 4 divided by (x - 1) gives x + 1 remainder -3. So y = x + 1 - 3/(x - 1), and the oblique asymptote is y = x + 1.

x-intercepts: x = 2 and x = -2. y-intercept: y = (-4)/(-1) = 4.

Graphical methods for solving equations

The specification requires you to solve equations, including transcendental functions (those involving sin, cos, e^x, ln, etc.), by graphical methods. The principle is straightforward: rearrange the equation so that one side is a function you can graph and the other side is another function you can graph. The solutions are the x-coordinates of the intersection points.

Worked example: graphical solution of a transcendental equation

Show that the equation e^x = 5 - 2x has a root between x = 0 and x = 1.

Let f(x) = e^x and g(x) = 5 - 2x.

At x = 0: f(0) = 1, g(0) = 5. Here f(0) < g(0).

At x = 1: f(1) = e = 2.718..., g(1) = 3. Here f(1) < g(1).

At x = 2: f(2) = e² = 7.389..., g(2) = 1. Here f(2) > g(2).

Since f is continuous and goes from being below g to above g between x = 1 and x = 2, there is a root in the interval (1, 2). (Checking more carefully: at x = 1.2, f(1.2) = 3.320..., g(1.2) = 2.6. So the root is between 1 and 1.2.)

Graphically, this is the point where the rising exponential curve y = e^x crosses the falling straight line y = 5 - 2x.

Worked example: using graph intersection to solve a trig equation

By sketching appropriate graphs, find the number of solutions of sin x = x/4 for 0 <= x <= 4pi.

Sketch y = sin x (oscillating between -1 and 1) and y = x/4 (a straight line through the origin with gradient 1/4). The line y = x/4 starts at the origin and reaches y = pi at x = 4pi, which is approximately 3.14. Since sin x oscillates between -1 and 1, the line exits the range of sin x after x = 4 (where x/4 = 1). The line crosses the sine curve at the origin and once more in the first positive arch of sin x, giving 2 solutions in (0, 4pi) where the line is below 1. After that, the line is above sin x. So there are 2 solutions total (including x = 0).

Practice questions

These edexcel igcse further pure mathematics practice questions cover the graphs section. Work through them with a pencil and graph paper.

1. Sketch the graph of y = (x - 1)(x + 2)(x - 3), labelling all axis intercepts.
2. Sketch the graph of y = (2x + 3) / (x - 1), labelling asymptotes and intercepts.
3. By sketching y = x² and y = 3 - 1/x on the same axes (for x > 0), determine the number of positive solutions to x² + 1/x = 3.
4. The curve y = (x² + 2x - 3) / (x + 1) has a vertical asymptote and an oblique asymptote. Find both and sketch the curve.

Solutions

Question 1: x-intercepts: x = 1, x = -2, x = 3 (all single roots). y-intercept: y = (-1)(2)(-3) = 6. Leading coefficient positive, degree 3, so the curve rises to the right and falls to the left. The sketch shows a cubic crossing at -2, 1 and 3, passing through (0, 6).

Question 2: Vertical asymptote: x = 1. Horizontal asymptote: y = 2. x-intercept: x = -3/2. y-intercept: y = -3. Two branches: left of x = 1 (below y = 2) and right of x = 1 (above y = 2).

Question 3: For x > 0, y = x² is a rising parabola and y = 3 - 1/x is a curve that rises from negative infinity (near x = 0) towards y = 3. They intersect once, so there is exactly 1 positive solution.

Question 4: x² + 2x - 3 = (x + 3)(x - 1). Vertical asymptote: x = -1. Divide: (x² + 2x - 3)/(x + 1) = x + 1 - 4/(x + 1). Oblique asymptote: y = x + 1. x-intercepts: x = -3 and x = 1. y-intercept: y = -3.

The igcse 4PM1 graphs topic is one where neat, precise presentation directly translates into marks. These edexcel igcse further pure mathematics notes should equip you with the systematic approach that the exam rewards: identify asymptotes, find intercepts, check end behaviour, and draw a clear sketch with all features labelled.

For additional edexcel igcse further pure mathematics explained content and practice across all specification areas, use edexcel igcse further pure mathematics revision notes on the Green Bridge CBT platform to sharpen your graphical skills and track your progress.