Graphs
Understanding Graphs

A graph in mathematics is a visual representation of a set of data or of a mathematical function.

This might be represented as a set of points, lines, curves, circles, or more complex figures.

In some cases, a graph might represent a mathematical function, showing the input and output for a specific rule.
Function Graphs

The graph of a function is the set of all points in the plane of the form (x, f(x)).

The function f defines the vertical displacement at any horizontal position x.

A function can have one graph, and one function graph can represent multiple functions.
Key Features of Graphs

Asymptotes are lines that the graph approaches but never crosses.

Intercepts are points where the graph crosses or touches the x or y axis.

The domain and range specify what values of x the graph is defined for, and which yvalues the graph takes.

The maximum and minimum values indicate the highest and lowest points of the graph.
Sketching Graphs

Sketching the graph of a function involves plotting a set of points that satisfy the function and then joining them.

Using table of values, x and y intercepts, asymptotes, and turning points may help create an accurate sketch.

The ability to sketch trigonometric, exponential, logarithmic, and polynomial function graphs is essential.
Transformations of Graphs

Transformations include translations, reflections, and stretching/scaling.

Translations shift a graph either vertically or horizontally.

Reflecting a graph flips it over either the x or y axis.

Stretching or scaling changes the size of the graph.
Parametric Equations

Parametric equations describe a graph or a path in terms of a third variable, often time, as well as x and y.

To sketch a parametric curve, plot points for different values of the parameter.
Implicit Functions

An implicit function is one which can’t be written as y = f(x), but in another form, often involving x and y on the same side of the equation.

You can sketch an implicit function by finding a series of points that satisfy the equation, and connecting them in a reasonable way.

An implicit function may produce graphs that don’t pass the vertical line test and therefore aren’t traditional functions.