This post discusses how the linear transformation rule for reflecting a figure over an oblique line y = mx + b can be used to create learning opportunities for high school students at different grade levels. Initially one might think that this topic is too advanced for most high school students. In my opinion, it’s possible to create a variety of activities based on linear transformations that will give mathematical nourishment to high school students at many different grade levels. An understanding of the concept of using a linear transformation to change a 2D graphic object to another 2D graphic object can definitely benefit college linear algebra students and computer graphic programmers. You may find it helpful to read my post, Geometric Transformations to Practice Basic Skills and Introduce Fundamental Concepts.

The content of this post is based on my free handout, Reflection over Any Oblique Line. This handout covers the following topics and items:

- The derivation of the liner transformation rule (p, q) → (r, s) that reflects a figure over the oblique line y = mx + b where both r and s are functions of p, q, m, and b.
- The derivation of the linear transformation rule (p, q) → (r, s) that reflects a figure over the oblique line y = mx + b where both r and s are functions of p, q, b, and θ = Tan
^{-1}(m). - Given the specific equation of a line y = mx + b, show different ways of finding a linear transformation rule to reflect a preimage figure over the line y = mx + b.
- Graphic images showing the reflection images of various polygons over different oblique lines.
- A 13-step algorithm for the TI-84 graphing calculator to draw preimage and image polygons under a linear transformation.

The linear transformation rule (p, s) → (r, s) for reflecting a figure over the oblique line y = mx + b where r and s are functions of p, q, m, and b is given below. Finding the linear transformation rule given equation y = mx + b involves substituting the values for m and b in the formulas below, and then using basic operations with fractions to simplify each of the six coefficients of the linear transformation rule. What a sneaky way to get kids to practice operations with fractions.

As shown in the handout, Reflection over Any Oblique Line, the linear transformation rule for reflecting over the line y = -2x + 4 is (p, q) → (-3/5p – 4/5q + 16/5, -4/5p + 3/5q + 8/5). When reflecting over the line y = 3/5x – 4, the linear transformation rule is (p, q) → (8/17p + 15/17q + 60/17, 15/17p – 8/17q – 100/17). The graph below shows the reflection images of polygons over the lines y = -2x + 4 and y = 3/5x – 4.

The linear transformation rule (p, s) → (r, s) for reflecting a figure over the oblique line y = mx + b where r and s are functions of p, q, b, and θ = Tan^{-1}(m) is shown below. Finding the linear transformation rule given the equation of the line of reflection equation y = mx + b involves using a calculator to find angle θ = Tan^{-1}(m), and then calculating each of the six coefficients of the linear transformation rule.

The handout, Reflection over Any Oblique Line, shows the derivations of the linear transformation rules for lines of reflection y = √(3)x – 4 and y = -4/5x + 4.

- Line y = √(3)x – 4: θ = Tan
^{-1}(√(3)) = 60° and b = -4. The corresponding linear transformation rule is (p, q) → (r, s) = (-0.5p + 0.866q + 3.464, 0.866p + 0.5q – 2). - Line y = -4/5x + 4: θ = Tan
^{-1}(-4/5) = -38.66° and b = 4. The corresponding linear transformation rule is (p, q) → (r, s) = (0.2195p – 0.9756q + 1.9512, -0.9765p – 0.2195q + 2.4390).

The graph below shows the reflection images of a polygon over the lines y = √(3)x – 4 and y = -4/5x + 4.

**Suggestions for activities that teachers might consider:**

- Give students a sheet of graph paper with the line of reflection and preimage polygon drawn. Also give them the equation of the line of reflection and the linear transformation rule corresponding to the equation of the line of reflection. Have students use the linear transformation rule to calculate the vertices of the image polygon, and then draw the image polygon. A completed assignment should include lists of the x-y coordinates of the vertices of the preimage and the image polygon.
- Give students a sheet of graph paper with the line of reflection and preimage polygon drawn. Have students find the equation of the line of reflection in slope-intercept format, and the linear transformation rule corresponding to the equation of the line of reflection. Depending on the level and interest of the students, allow students to calculate transformation rule coefficients in terms of parameters m and b or θ and b. Then have students use the linear transformation rule to calculate the vertices of the image polygon and draw the image polygon. A completed assignment should include lists of the x-y coordinates of the vertices of the preimage and the image polygon.
- Have upper level students derive the linear transformation rule with parameters m and b. As illustrated in the handout, Reflection over Any Oblique Line, explain and discuss the general strategy for deriving the linear transformation rule with parameter m and b. Derivation of the equations for a linear transformation only requires an understanding of concepts already encountered in their math classes. This is a great activity for promoting mathematical reasoning and presenting ideas in an organized manner to explain mathematical relationships.
- Have upper level trig students derive the linear transformation rule with parameters θ and b. Using the handout, Reflection over Any Oblique Line, as a guide, review and discuss the trig identities needed to convert the six coefficients of a linear transformation with parameters m and b to the coefficients of a linear transformation with parameters θ and b. Students now have a need to use trig identities.
- If students have access to the appropriate technology, have them do a project in which they graph the reflection image of a polygon over an oblique line y = mx + b. The completed project should include the following items: 1) A graph showing the line of reflection, preimage polygon, and image polygon. 2) The equation of the line of reflection. 3) A describtion of the linear transformation rule corresponding to the equation of the line of reflection. 4) A list of the x-y coordinates of the vertices of the preimage polygon. 5) A list of the x-y coordinates of the vertices of the image polygon. Students who have access to a TI-84 graphing calculator can use the 13-step algorithm given in the handout Reflection over Any Oblique Line.
- The handout, Reflection over Any Oblique Line, shows how linear transformation rules for reflections over lines can be expressed in terms of matrix multiplication. After showing students matrix multiplication based transformation rules, they better understand why matrix multiplication is done the way it is. Programmers use matrix multiplication to perform 2D and 3D transformations of objects on a computer screen. Computer video cards are optimized to perform millions of matrix multiplications per second.
- Recruit a team of computer programming geeks in your school to write a program that calculates the x-y coordinates of a reflection image of a figure over any oblique line y = mx + b, and then graph the line of reflection, the preimage figure, and the image figure. The set of features the program could offer is limited only by the ability and imagination of the programmers.