Relations, Functions and Graphs 2

Section 1Non-linear graphs

A non-linear graph is not a straight line. Quadratic graphs form parabolas, inverse variation graphs curve toward the axes, and other relationships may bend or change steepness. Watch out: a table with a constant first difference gives a straight line; a changing first difference suggests a curve.

Section 2Quadratic graphs

A quadratic graph has an x^2 term and is shaped like a parabola. The graph of y = x^2 opens upwards, has a minimum at the origin, and is symmetrical about the y-axis. In plain words, equal positive and negative x-values give the same y-value.

Section 3Graph transformations

Transformations change a graph's position or shape. Adding outside the function moves the graph up or down, such as y = x^2 + 3 moving y = x^2 up 3 units. Replacing x with x - a moves the graph right a units, which is easy to mix up because the sign looks backwards.

Section 4Reflections and stretches

A negative outside a function reflects the graph in the x-axis, such as y = -x^2. Multiplying the whole function by a number greater than 1 makes the graph vertically steeper. Multiplying by a fraction between 0 and 1 makes it flatter.

Section 5Direct variation

Direct variation means one quantity is a constant multiple of another: y = kx. The graph is a straight line through the origin. Here's the trick: y/x stays constant for direct variation.

Section 6Inverse variation

Inverse variation means one quantity increases while the other decreases so their product stays constant: y = k/x. The graph is a curve that gets close to the axes but does not touch them. For example, if workers share a fixed job, more workers usually means fewer hours per worker.

Section 7Interpreting transformed graphs

When interpreting a graph, connect the shape to the real situation. A taxi fare graph may be linear, but the area of a square against side length is quadratic. A trip from Kingston to Montego Bay at fixed distance can show inverse variation between speed and time.