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 y-values 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.