# Translating Lines

## Problem

*Translating Lines card matching activity*

Move the red dots to set the position of the red line. Then move the blue dot to translate the blue line up and down.

Investigate what happens to the equations of different lines when you translate them up or down.

Try to predict what will happen. Explain your findings.

Now investigate what happens to the equations of different lines when you translate them left or right.

Try to predict what will happen. Explain your findings.

Below are the equations of ten lines, and some translations. Each translation links a pair of parallel lines - can you match them up?

$y=2x-5$ |
$y=-2x+6$ | Right 4 |

$y=\frac13x+24$ |
$y=2x+12$ | Right 5, Up 5 |

$y=-\frac13x-11$ |
$y=\frac13x+21$ | Left 1, Up 2 |

$y=-2x-9$ |
$y=2x-9$ | Left 6, Down 5 |

$y=2x+4$ |
$y=-\frac13x-12$ | Left 3 |

Can you find other combinations of vertical and horizontal translations that link each pair of parallel lines?

## 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:

You can tell it's translated because the gradient hasn't changed, and the pairs of lines are parallel.

Ellie from Chiswick expanded and gave reasoning;

I made this into a formula.

If the original line is $y=ax+b$, and we move $X$ right and $Y$ up, then the new line is $y=ax+b-aX+Y$. So the constant becomes $b-aX+Y$.

So the constant is the same, but minus the gradient times the movement (if you move left then $X$ is negative), and then plus the vertical movement (of you move down then $Y$ is negative).

So the pairs are:

So lots of values of $X$ and $Y$ will make these work, as you can make equations that describe the motion, and then any solutions will work. You could even plot the solutions, they themselves would be straight line graphs.

So for $y=2x+1$ translated to $y=2x-4$, the $X$ and $Y$ motion satisfies $1-2X+Y=-4$ by the equation above, so any $X,Y$ satisfying $Y-2X=-5$ will do the translation above.

## 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 of translating a straight line on the equation that represents or defines it. 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 Reflecting Lines.

Working with the whole group demonstrate the interactivity by lining up the dots so the two lines are the same. Show how the line can be translated vertically by moving the blue dot. Draw attention to the equations of the lines showing next to the graph.

Choose a suitable line, and tell the students you are going to translate it up or down by a number of units. Ask them to picture what this will look like in order to predict what the equation of the new line will be. Do this a number of times with different lines and translations until students are able to predict the new equation with confidence. Ask them to share insights and explanations/justifications.

The second part of this problem is perhaps a little more challenging. Demonstrate, using the interactivity, how the line can be translated horizontally by moving the blue dot. Give students plenty of time, perhaps working in pairs at computers, to picture and sketch the effect these translations 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 translated horizontally by a given number of units.

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 translations 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 and suggest the students work on this in pairs, with the aim of producing a display of their results. This could include sketches of the graphs and suggestions of other combined translations which have the same outcome.

### Key questions

When we translate a graph, what changes? What stays the same?

What information do we need to predict what will happen to the intercept when we translate horizontally?

For fractional gradients - how far do lines need to be translated horizontally for the intercept to change by a whole number?

### 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.

Then structure the activity above so that students start by working on simpler cases.

### Possible extension

**Instead of** the card matching activity above, students could do the same activity with this set of cards which has six additional lines (all with fractional gradients) and three additional translation cards.

Students could create their own card matching activity for their peers to complete.

An interesting line of enquiry is to look at pairs of translations (one vertical, one horizontal) which link a pair of parallel lines. Challenge students to explain why there are an infinite number of possibilities.

Another option is to explore gradients of lines and the corresponding horizontal translations which lead to a given change in intercept. Can they explain any relationships they find?