The ability to apply the equation transformation rules is one of the most important skills that math students can learn. When they recognize that a given graph is just a geometric transformation of the graph of some familiar basic equation, it’s relatively easy for them to find the equation of the graph. Likewise, when they recognize that a given equation is just an algebraically transformed familiar basic equation, it’s relatively easy for them to draw a sketch of the graph.

Most functions and relations beginning algebra through calculus students encounter are the result of applying an ordered series of algebraic transformations to a basic equation. Each algebraic transformation applied results in a geometric transformation of the equation’s graph. A set of basic equation transformation rules describes how graphs can be translated, reflected, and rotated.

In this post, we will see how the equation transformation rules can be used to transform the graph of the square root function. To best understand this demonstration, download the free handout *Equation Transformation Rules* from www.mathteachersresource.com/instructional-content.html. It contains a succinct summary of the equation transformation rules, a simple explanation of why some of the counterintuitive rules work, and examples that show how the transformation rules can be applied. This handout is a helpful resource for both students and teachers.

Teaching Points:

- The biggest mistake students make is replacing x with x + k or x – k. Remind students to enclose x + k or x – k in parentheses and then simplify the equation.

- When reflecting the graph over the y-axis, replace every x with (-x) and then simplify the equation.

- When finding the equation of a given graph, results should be checked by picking a few key points on the given graph and then determine whether or not the x-y coordinates of these key points fit the equation.

- When finding the equation of a graph, teach students how they should see a graph.

Example 1: I see a line with a slope of 3 that has been slide horizontally 7 units to the right.

Example 2: I see a cosine curve, amplitude of 4, that has been flipped over the x-axis.

Example 3: I see a circle, radius of 4, that has been stretched vertically by a factor of 5/2.

- Depending on the course content, students should be required to memorize the equation, the shape, and the properties of basic functions and relations. Students can make flash cards.

The above graphs, created with the program Basic Trig Functions, are offered by Math Teacher’s Resource. The program has features that facilitate learning and teaching the equation transformation rules. You can enter an equation in any of the formats shown in the examples above. Except for exponents, all equations are entered like any equation in a textbook. For example: The inequality 2x – 10Sin^{3}(3x) + 4y^{2} ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. You can specify whether to shade the intersection or union of a system of inequalities.

The user interface, simple and intuitive for all program modules, provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, you can find relative minimum points, relative maximum points, x-intercepts and intersection points with simple mouse control clicks. A Help menu provides a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.