# Graphs of Functions and Algebra - Interactive Tutorials

Free tutorials to explore important topics in precalculus such as quadratic, rational, exponential, logarithmic, trigonometric, polynomial, absolute value functions and their graphs are included. Equations of lines, circles, ellipses, hyperbolas, and parabolas are also explored interactively. Graph shifting, scaling, and reflection are also included. The definition and properties of inverse functions are thoroughly investigated. A graphical approach to 2 by 2 systems of equations is included.

## Functions

• Questions on Functions (with Solutions). Several questions on functions are presented and their detailed solutions are discussed.
• Operations on Functions are presented through examples and questions with solutions.
• Linear Functions. A tutorial to explore the graphs, domains and ranges of linear functions.
• Square Root Functions. Square root functions of the form f(x) = a √(x - c) + d and the characteristics of their graphs such as domain, range, x intercept, y intercept are explored interactively.
• Cube Root Functions. Cube root functions of the form f(x) = a (x - c) 1/3 + d and the properties of their graphs such as domain, range, x intercept, y intercept are explored interactively using an applet.
• Cubing Functions. Graphs of the cubing functions of the form f(x) = a (x - c) 3 + d as well as their properties such as domain, range, x intercept, y intercept are explored interactively using an applet.
• Graph, Domain and Range of Common Functions. A tutorial using a large window applet to explore the graphs, domains and ranges of some of the most common functions used in mathematics.
• Quadratic Functions (general form). Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an applet.
• Quadratic Functions(standard form). Quadratic functions in standard form f(x) = a(x - h) 2 + k and the properties of their graphs such as vertex and x and y intercepts are explored, interactively, using an applet.
• The Product of Two Linear Functions Gives a Quadratic Function. This property is explored interactively using an applet.
• Even and Odd Functions. Graphical and analytical examples on even and odd functions.
• Periodic Functions. Graphical and analytical examples with solutions of periodic functions.
• Absolute Value Functions. Absolute value functions definition and graph are explored, using an HTML5 app, by comparing the graphs of f(x) and h(x) = |f(x)|.

## Hyperbolic Functions

• Graphs of Hyperbolic Functions. The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are explored using an applet.

## Graph Transformations

• Vertical Stretching and Compression. This applet helps you explore, interactively, and understand the stretching and compression of the graph of a function when this function is multiplied by a constant a.
• Reflection of Graphs In x-axis. This is an applet to explore the reflection of graphs in the x-axis by comparing the graphs of f(x) (in blue) and h(x) = -f(x) (in red).
• Reflection of Graphs In y-axis. This is an applet to explore the reflection of graphs in the y-axis by comparing the graphs of f(x)(in blue) and h(x) = f(-x) (in red).
• Reflection Of Graphs Of Functions. This is an applet to explore the reflection of graphs in the y axis and x axes. Graphs of f(x), f(-x), -f(-x) and -f(x) are compared and discussed.

## Equation of Line

• Find Equation of a Line from a Graph, examples with detailed solutions.
• Slope of a Line. The slope of a straight line, parallel and perpendicular lines are all explored interactively using an applet.
• General Equation of a Line: ax + by = c. Explore the graph of the general linear equation in two variables that has the form ax + by = cmthrough examples.
• Slope Intercept Form Of The Equation Of a Line. The slope intercept form of the equation of a line is explored interactively using an applet. The investigation is carried out by changing parameters m and b in the equation of a line given by y = mx + b.
• Find Equation of a Line - applet. An applet that generates two lines. One in blue that you can control by changing parameters m (slope) and b (y-intercept). The second line is the red one and it is generated randomly. As an exercise, you need to find an equation to the red line of the slope intercept form y = mx + b.

## Equation of Circle

• Equation of a Circle. An applet to explore the equation of a circle and the properties of the circle. The equation used is the standard equation that has the form (x - h) 2 + (y - k) 2 = r 2.
• Find Equation of Circle - applet. This is an applet that generates two graphs of circles. The equations of these circles are of the form (x - h) 2 + (y - k) 2 = r 2. You can control the parameters of the blue circle by changing parameters h, k and r. The second circle is the red one and it is generated randomly. As an exercise, you need to find an equation to the red circle.

## Equation of Ellipse

• Equation of an Ellipse. This is an applet to explore the properties of the ellipse given by the following equation (x - h) 2 / a 2 + (y - k) 2 / b 2 = 1.

## Equation of Hyperbola

• Equation of Hyperbola. The equation and properties of a hyperbola are explored interactively using an applet. The equation used has the form x 2/a 2 - y 2/b 2 = 1 where a and b are positive real numbers.

## Polar Coordinates And Equations

• Polar Coordinates and Equations. The graphs of some specific polar equations are explored using java applet. You can also plot your own points generated using the polar equation under investigation.