# Transformations and Coordinates

In this lesson, we will use coordinates to study similarity transformations (translations, reflections, rotations, and dilations). You will learn a number of simple formulas for the results of these transformations, using the coordinates of the points being transformed.

## Formulas for translations

A red point \$(x_1,y_1)\$ is plotted on the grid to the left. You can use the sliders below the grid to move the red point around. The blue point \$(x_2,y_2)\$ plotted to the left is the result of translating the red point 3 units to the right and 2 units up; as you move the red point, the blue point moves along with it.

For each row in the table below, slide the sliders to move the red point to the indicated coordinates. Then give the coordinates of the blue point.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(x_2,y_2)}\$

The table below gives you all the red points \$(x_1,y_1)\$ from transformQn1. They are also plotted on the grid to the left, and connected with line segments. For each of these points, type in the point \$(x_1+3,y_1+2)\$, which will be plotted in blue.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(x_1+3,y_1+2)}\$
 Are the blue points from the table in this question in the same locations as the blue points from the table in transformQn1?
 If a point \$(x_2,y_2)\$ is the result of translating a point \$(x_1,y_1)\$ 3 units to the right and 2 units up, what is a formula for \$(x_2,y_2)\$?

For each row in the table below, a red shape will be drawn on the top grid to the left, as well as some translation of it in blue. In each row, click the button to see the shapes, and then give a formula for the translation shown on the top grid. The result of applying your formula to the red shape will be shown on the bottom grid to the left. (This means that, when you have the right answer, the pictures on the two grids will match.)

TranslationFormula

As you can see:

If a point \$(x_2,y_2)\$ is the result of translating a point \$(x_1,y_1)\$ to the right by \$a\$ units and up by \$b\$ units, then \$(x_2,y_2)=(x_1+a,y_1+b)\$.

## Formulas for reflections

A red point \$(x_1,y_1)\$ is plotted on the grid to the left. You can use the sliders below the grid to move the red point around. The blue point \$(x_2,y_2)\$ plotted to the left is the result of reflecting the red point across the \$y\$-axis.

For each row in the table below, slide the sliders to move the red point to the indicated coordinates. Then give the coordinates of the blue point.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(x_2,y_2)}\$

As you can see:

If a point \$(x_2,y_2)\$ is the result of reflecting a point \$(x_1,y_1)\$ across the \$y\$-axis, then \$(x_2,y_2)=(-x_1,y_1)\$.

A red point \$(x_1,y_1)\$ and sliders to move it are shown to the left, as before. Now the blue point \$(x_2,y_2)\$ plotted to the left is the result of reflecting the red point across the \$x\$-axis.

For each row in the table below, give the coordinates of the blue point which corresponds to the given red point.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(x_2,y_2)}\$

As you can see:

If a point \$(x_2,y_2)\$ is the result of reflecting a point \$(x_1,y_1)\$ across the \$x\$-axis, then \$(x_2,y_2)=(x_1,-y_1)\$.

## Formulas for rotations

The top grid to the left shows a red zigzag and a blue zigzag. The blue zigzag is the result of rotating the red zigzag around the origin by 90˚ (counterclockwise).

Type the coordinates of each blue point into the table below, in the order given by the line segments connecting them. The result can be seen on the bottom grid to the left.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(x_2,y_2)}\$
 Look at the second row of the table, where \$\cl"red"{(x_1,y_1)}=(1,2)\$. For these values of \$x_1\$ and \$y_1\$, what is \$(-y_1,x_1)\$?
 Does that match the value of \$\cl"blue"{(x_2,y_2)}\$ in that row?
 Look at the rest of the table. For each point \$\cl"red"{(x_1,y_1)}\$ in the first column, is the value of \$(-y_1,x_1)\$ equal to the value of \$\cl"blue"{(x_2,y_2)}\$ in the second column?

As you can see:

If a point \$(x_2,y_2)\$ is the result of rotating a point \$(x_1,y_1)\$ around the origin by 90˚ (counterclockwise), then \$(x_2,y_2)=(-y_1,x_1)\$.

The top grid to the left shows a red zigzag and a blue zigzag. Now the blue zigzag is the result of rotating the red zigzag around the origin by 180˚.

Type the coordinates of each blue point into the table below, in the order given by the line segments connecting them. The result can be seen on the bottom grid to the left.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(x_2,y_2)}\$
 Look at the second row of the table, where \$\cl"red"{(x_1,y_1)}=(1,2)\$. For these values of \$x_1\$ and \$y_1\$, what is \$(-x_1,-y_1)\$?
 Does that match the value of \$\cl"blue"{(x_2,y_2)}\$ in that row?
 Look at the rest of the table. For each point \$\cl"red"{(x_1,y_1)}\$ in the first column, is the value of \$(-x_1,-y_1)\$ equal to the value of \$\cl"blue"{(x_2,y_2)}\$ in the second column?

As you can see:

If a point \$(x_2,y_2)\$ is the result of rotating a point \$(x_1,y_1)\$ around the origin by 180˚, then \$(x_2,y_2)=(-x_1,-y_1)\$.

The top grid to the left shows a red Z, as well as the blue Z which is the result of rotating the red Z around the origin by \$-90˚\$ (that is, clockwise by 90˚).

You can find a formula for rotating by \$-90˚\$ in the same way as we found formulas for rotating by 90˚ and 180˚ in the last two questions. It turns out that:

If a point \$(x_2,y_2)\$ is the result of rotating a point \$(x_1,y_1)\$ around the origin by \$-90˚\$ (that is, 90˚ clockwise), then \$(x_2,y_2)=(y_1,-x_1)\$.

For each point \$\cl"red"{(x_1,y_1)}\$ in the red Z, type the point \$\cl"blue"{(y_1,-x_1)}\$ into the table below. The result will be shown on the bottom grid to the left.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(y_1,-x_1)}\$
 After you have filled in the table, does the bottom grid look identical to the top grid?

## Formulas for dilations

The red triangle drawn on the top grid to the left connects the point \$\cl"red"{(x_1,y_1)}\$ to the origin. Notice that one side runs along the \$x\$-axis, and another side is parallel to the \$y\$-axis.

 What is the length of the side of the triangle that runs along the \$x\$-axis?
 What is the length of the side of the triangle that is parallel to the \$y\$-axis?
 What are the coordinates of the point \$\cl"red"{(x_1,y_1)}\$?

The blue triangle drawn on the bottom grid to the left is a dilation of the red triangle about the origin by \$r\$. The point \$\cl"blue"{(x_2,y_2)}\$ is the dilation of \$\cl"red"{(x_1,y_1)}\$.

 When \$r=2\$, what is the length of the side of the blue triangle that runs along the \$x\$-axis?
 When \$r=2\$, what is the length of the side of the blue triangle that is parallel to the \$y\$-axis?
 When \$r=2\$, what are the coordinates of the point \$\cl"blue"{(x_2,y_2)}\$?
 As you slide the \$r\$ slider, notice what happens to the coordinates of the point \$\cl"blue"{(x_2,y_2)}\$. Can you always get them by looking at the sides of the blue triangle?

Dilating a triangle by \$r\$ multiplies the lengths of all its sides by \$r\$. So dilating the red triangle by \$r\$ gives a triangle where the side that runs along the \$x\$-axis has length \$2r\$, and the side that is parallel to the \$y\$-axis has length \$3r\$.

 Give a formula for the coordinates of the point \$\cl"blue"{(x_2,y_2)}\$.

You can use this triangle method to see that:

For any \$r>0\$, if you dilate a point about the origin by \$r\$, its coordinates can be obtained by multiplying the original point’s coordinates by \$r\$. That is, the dilation of the point \$(x,y)\$ about the origin by \$r\$ is the point \$(rx, ry)\$.

The table below gives you a collection of red points \$(x_1,y_1)\$. They are also plotted on the grid to the left, and connected with line segments. For each of these points, type in its dilation about the origin by 2, \$(2x_1,2y_1)\$. These dilated points will be plotted in blue.

\$\cl"red"{(x_1,y_1)}\$\$\cl"blue"{(2x_1,2y_1)}\$