Harder Graphs

Dealing with Cubic Graphs

Cubic graphs, or graphs representing cubic functions, are an important part of algebra and are recognisable by their distinct ‘S’ shaped curve.
Cubic graphs are particularly useful in modelling real-world situations where the rate of change is not constant.
The general equation is y = ax^3 + bx^2 + cx + d, where ‘a’ is crucial as it defines the direction of the curve (positive ‘a’ for right-hand up and negative ‘a’ for right-hand down).
The points at which these graphs cross the x-axis are called the roots of the cubic equation.

Recognising and Sketching Quadratic Graphs

A quadratic graph represents a quadratic function and is a U-shaped curve known as a parabola.
The general form of a quadratic equation is y = ax^2 + bx + c.
The vertex of the parabola is the peak if the graph opens downwards and the lowest point if the graph opens upwards. It can be found by the formula, (-b/2a, f(-b/2a)).
If the graph cuts through the x-axis, these points are known as the roots of the quadratic equation.

Exploring Reciprocal Graphs

A reciprocal graph represents reciprocal functions and has a characteristic ‘horseshoe’ shape.
The general form is y = 1/x or y = k/x.
These graphs have two key features: an asymptote on the x-axis and an asymptote on the y-axis.
No part of the graph ever touches the asymptotes, which are lines the graph shape gets indefinitely close to without ever reaching.

Adjustments to Graphs

Transformations can be applied to graphs including translations, reflections, stretch and compressions.
Translations move the graph horizontally or vertically. For example, y = x^2 + 3 shifts the graph of y = x^2 three units upward.
Reflections flip the graph. When y = -f(x), the graph of f(x) is flipped around the x-axis.
Stretching and compression change the shape of the graph. For example, y = 2f(x) vertically stretches the graph of f(x) by a factor of 2.

Importance of Further Practice

Experimenting with different equations and adjusting the constants allows for a better understanding of how these changes affect the graph.
Practising sketching these graphs freehand can improve intuition about the shapes and patterns of these functions.
Understanding these graphs and transformations is essential for solving and interpreting more complex algebraic problems.

Remember to always consider the real-world context of problems, as this can often help in understanding what the graph is showing and why certain features are important.