Harder Graphs (AQA GCSE Maths): Revision Notes

Harder graphs

Understanding different types of harder graphs is essential for GCSE mathematics. This revision note covers the main types of harder graphs you'll encounter, including their equations, key properties, and how to sketch them effectively.

Cubic graphs (x³ graphs)

Cubic graphs follow the general form $y = ax³ + bx² + cx + d$, where any of the coefficients b, c, or d can be zero. These graphs are also known as polynomial functions and have some distinctive characteristics that make them recognisable.

The most important feature of cubic graphs is their characteristic shape - they always have a "wiggle" in the middle. This wiggle can be quite flat or more pronounced depending on the specific equation. The key thing to remember is that x³ must be the highest power in the equation, and you cannot have any fractional terms like 1/x.

When sketching cubic graphs, the direction they travel depends on the coefficient of x³. If the coefficient is positive (like +x³), the graph goes up from bottom left to top right. If it's negative (like -x³), the graph goes down from top left to bottom right.

Circle equations and tangent lines

Circles centred at the origin follow a very specific equation format. When a circle has its centre at the point (0, 0) and radius r, its equation is $x² + y² = r²$. This is a fundamental relationship that you must memorise.

For example, the equation $x² + y² = 25$ represents a circle with centre (0, 0) and radius 5, because $r² = 25$, so $r = 5$. Similarly, $x² + y² = 100$ represents a circle with centre (0, 0) and radius 10.

Finding the equation of a tangent line to a circle involves understanding the relationship between the radius and the tangent. A tangent line always meets the radius at 90°, meaning they are perpendicular to each other. This perpendicular relationship is the key to solving tangent problems.

Exponential graphs (k^x graphs)

Exponential graphs come in two main forms: $y = k^x$ and $y = k^{-x}$, where k is a positive number. These graphs represent exponential growth and exponential decay respectively, and they have several important properties you need to understand.

All exponential graphs share certain characteristics: they are always positioned above the x-axis, and they always pass through the point (0, 1). This happens because any number raised to the power of 0 equals 1.

For exponential growth (when the power is positive), if k is greater than 1, the graph curves upwards as x increases. For exponential decay (when the power is negative), the graph is flipped horizontally and curves downwards as x increases, approaching the x-axis but never actually touching it.

Reciprocal graphs (1/x graphs)

Reciprocal graphs follow the forms $y = \frac{A}{x}$ or $xy = A$, where A is a constant. These graphs create a distinctive hyperbola shape and have some unique properties that set them apart from other graph types.

The graph consists of two separate curved sections that never actually touch each other.

All reciprocal graphs with the same basic equation have the same fundamental shape, but their orientation depends on whether the constant A is positive or negative. Positive values create curves in the first and third quadrants, while negative values create curves in the second and fourth quadrants.

These graphs are symmetrical about the lines y = x and y = -x. The two halves of the hyperbola mirror each other across these diagonal lines. This symmetry is a useful property when sketching reciprocal graphs.

Trigonometric graphs

Trigonometric graphs include sine, cosine, and tangent functions, each with their own distinctive shapes and properties. Understanding these graphs is crucial for solving trigonometric problems and recognising periodic patterns.

Sine and cosine graphs have very similar underlying shapes - they both oscillate between upper and lower limits of exactly +1 and -1. The main difference is that the sine graph is shifted 90° to the right compared to the cosine graph. People often describe sine graphs as "waves" and cosine graphs as "buckets" because of their visual appearance.

The tangent function is quite different from sine and cosine. Unlike the other trigonometric functions, tangent can take any value from negative infinity to positive infinity. It has a completely different shape, with vertical asymptotes (undefined points) at ±90°, ±270°, and so on.

Tangent repeats every 180°, which is half the period of sine and cosine. The graph consists of separate curved sections between each pair of asymptotes. When sketching tangent graphs, the easiest approach is to plot the key points that occur every 90° and then connect them with smooth curves.