Quadratic and Cubic Transformations

This applet investigates and compares transformations of cubic and quadratic functions and graphs in the form
y = A(Bx + C)³+ D or y = A(Bx + C)²+ D.

Author and programmer: Ron Barrow

UK Years 12-13, Ages 16-19, KS5, Core AS and A2 Mathematics - Functions and Graphs
US Grades 11, 12 - Algebra I and Algebra II


Instructions below   See also:  Polynomial Graphs   Solving Cubic Equations

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How to Use this Applet

The applet on this page is designed to help you understand and visualize transformations of simple polynomials - quadratics and cubics.
Cubics have the form: A(Bx + C)3 + D. [Notice that not all possible cubic expressions can be represented like this.]
Quadratics have the form: A(Bx + C)2 + D.
[This is the completed square form of the quadratic. Unlike cubics, any quadratic can be written in this form.]
You should investigate the effect of varying the parameters A, B, C and D. The instructions are easy to guess at, so I'll let you get on with it!
To test your understanding, hitting the "Random Curve" button will generate a new random curve.

By varying the parameters, try to cover the random line with the other one, and hence find the equation of the random curve. There is sometimes more than one correct answer.
If you want an easier start, then clicking on "1" at the top will set the program so that only one of A, B, C, D changes on the random curve - the others stay the same. It gets more complicated as you click "2", "3" and "4"! Have a go.
If the "Easier" box is ticked, then the value of B in the random curve is never negative, which makes things a little easier. Clear the tick if you want to allow negative values of B in the random curve - a trickier challenge.
Have some fun! Studying this function should also help understand functions and their transformations in general.