NRICH: Reflecting Lines

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NRICH: Reflecting Lines

Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.

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.

screenshot of GeoGebra tool showing two lines reflected in the line y=0 screenshot of GeoGebra tool, showing two lines reflected in the line x=0.

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?

Below are the equations of sixteen straight lines. Each line has a partner, either its reflection in the horizontal axis or its reflection in the vertical axis. Without plotting any graphs, can you find all the pairs and say which axis they were reflected in?
y=3x+14
 y=4x+3 y=2x+4 y=2x+14
y=−14x+3 y=−4x−2 y=14x+2 y=−2x−4
y=−14x+2 y=3x+4 y=−3x−14 y=14x−3
y=−2x+14 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.Age 11 to 14

NRICH (University of Cambridge)
Math Topics
Algebra & Pre-Algebra, Functions & Graphs, Linear Equations
Middle School, High School, Educator
6th Grade, 7th Grade, 8th Grade, 9th Grade
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NRICH: Reflecting Lines

Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.

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.

screenshot of GeoGebra tool showing two lines reflected in the line y=0 screenshot of GeoGebra tool, showing two lines reflected in the line x=0.

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?

Below are the equations of sixteen straight lines. Each line has a partner, either its reflection in the horizontal axis or its reflection in the vertical axis. Without plotting any graphs, can you find all the pairs and say which axis they were reflected in?
y=3x+14
 y=4x+3 y=2x+4 y=2x+14
y=−14x+3 y=−4x−2 y=14x+2 y=−2x−4
y=−14x+2 y=3x+4 y=−3x−14 y=14x−3
y=−2x+14 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.Age 11 to 14

NRICH (University of Cambridge)
Math Topics
Algebra & Pre-Algebra, Functions & Graphs, Linear Equations
Middle School, High School, Educator
6th Grade, 7th Grade, 8th Grade, 9th Grade

What are you looking for?

Organization

NRICH (University of Cambridge)

Website URL

Type of Resource

Challenge
Online Interactive Activity

Assigned Categories