Reflecting lines
Problem
Reflecting Lines printable sheet
Here are two sets of graphs. Each shows a pair of lines which are reflections of each other, one in the horizontal axis and one in the vertical axis.
Move the red and blue dots on the interactivity below to create some more pairs of reflected lines.
What can you say about the equations of two lines if one is a reflection of the other in the horizontal axis? What about a reflection in the vertical axis?
|
$y=3x+\frac14$ |
$y=4x+3$ | $y=2x+4$ | $y=2x+\frac14$ |
$y=-\frac14x+3$ |
$y=-4x-2$ | $y=\frac14x+2$ | $y=-2x-4$ |
$y=-\frac14x+2$ |
$y=3x+4$ | $y=-3x-\frac14$ | $y=\frac14x-3$ |
$y=-2x+\frac14$ |
$y=-4x+3$ | $y=4x+2$ | $y=-3x+4$ |
Now imagine that a line is reflected in one of the axes and the image is then reflected in the other.
Can you predict the equation of the resulting line if you know the equation of the original?
Does it make a difference which axis you choose to reflect in first?
Explain your findings.
Getting Started
You may want to have a look at Parallel Lines before working on this problem.
Why not start by investigating some equations whose graphs you can visualise easily?
Student Solutions
Jiwon Jung noticed:
The lines that are reflections have the same absolute value of their x coefficient and the constant.
Well noticed. Ellie from Chiswick matched up the reflections, and explained why they work:
Teachers' Resources
Why do this problem?
This problem helps students to consolidate their understanding of how equations of the form $y=mx+c$ describe the gradient and position of lines. Students explore the effect on the equation of reflecting a straight line. Students are encouraged to visualise the movement of the graphs in order to conjecture and test their conjectures.
This is good preparation for future work on transforming the graphs of more complicated functions.
Possible approach
It is useful if students have done some preliminary work plotting straight line graphs. The problems Parallel Lines and Diamond Collector offer good starting points. This problem works well alongside Translating Lines.
Working with the whole group, demonstrate the interactivity. Decide on the position of one of the lines. Ask the class to describe how to create the reflection of the line in each of the axes. How can they be sure they have reflected correctly?
Give students plenty of time, perhaps working in pairs at computers, to explore the effect that reflections have on the equations of lines. Clarify to the students that ultimately, the challenge is to be able to predict the new equation whenever a straight line is reflected in one of the axes.
Later, bring the class together and use the interactivity to test their ability to do this. Do this a number of times with different lines, and reflections in either axis, until students are able to predict the new equation with confidence. Ask them to share insights and explanations/justifications.
Hand out this card matching activity to check that students are confident in pairing reflected graphs and identifying the line of symmetry. If students need more practice they can create their own sets of cards for their peers to pair off.
Introduce the final part of the problem: can students predict the equation of the resulting line when an original line is reflected in one axis and the image is then reflected in the other axis? Expect them to explain and justify their findings, perhaps by producing a poster for display or making a presentation to the class.
Key questions
When we reflect a graph, what changes? What stays the same?
Possible support
Ensure that students are secure about the relationship between a line's properties and its equation. Encourage students to sketch the graphs of different equations and then use the interactivity to test their predictions.
Possible extension
Choose a straight line that doesn't go through the origin. Reflect in one of the axes. Reflect both lines in the other axis.
Students could now try the problem Surprising Transformations