Reflection

Worked example

1. 1

   $(4,1)$ — Take the first corner. Its x is 4 and its y is 1.

2. 2

   $(1,4)$ — Swap the two numbers. The line $y=x$ sends each x to the y place and each y to the x place.

3. 3

   $(6,1)\to (1,6)$ — Second corner: swap again. 6 and 1 trade places.

4. 4

   $(6,3)\to (3,6)$ — Third corner: swap again.

5. 5

   $(2,2)\to (2,2)$ — A point with x equal to y is already on the line $y=x$, so it does not move.

6. 6

   $\left(\frac{4+1}{2},\frac{1+4}{2}\right)=(2.5,2.5)$ — Check: the point halfway between (4, 1) and its image (1, 4) has x equal to y, so it is on the mirror line.

Worked example

1. 1

   $(2,1)\text{and}x=5$ — Take the corner (2, 1). The line $x=5$ is upright, so only the x number changes. The y stays 1.

2. 2

   $5-2=3$ — Find the distance from the corner to the line: 3 squares across.

3. 3

   $5+3=8$ — Go the same distance again, on the other side of the line. That is to the right, so add.

4. 4

   $(2,1)\to (8,1)$ — Write the image. The new x is 8 and the y is still 1.

5. 5

   $5-3=2$ — Next corner (3, 1): its distance from the line is 2 squares.

6. 6

   $5+2=7,(3,1)\to (7,1)$ — Go 2 past the line, so the new x is 7.

7. 7

   $(2,4)\to (8,4)$ — The last corner (2, 4) has the same x as the first, so its new x is also 8. The y stays 4.

Worked example

1. 1

   $(2,3)\text{and}x=-1$ — Now a line on the negative side. It is still upright, so only the x number changes.

2. 2

   $2-(-1)=3$ — Distance from the line to the corner: take away the line's x. Use brackets for the negative. The corner is 3 to the right.

3. 3

   $-1-3=-4$ — Go the same distance again, to the left of the line, so take 3 away from $-1$.

4. 4

   $(2,3)\to (-4,3)$ — Write the image. The y stays 3.

5. 5

   $-1-(-3)=2$ — Another corner, $(-3,2)$, is on the left of the line. Its distance is 2.

6. 6

   $-1+2=1,(-3,2)\to (1,2)$ — Go 2 past the line to the right, so add 2 to $-1$.

Worked example

1. 1

   $(2,1)\text{and}y=-1$ — Take the corner (2, 1). The line $y=-1$ is flat, so only the y number changes. The x stays 2.

2. 2

   $1-(-1)=2$ — Distance from the line to the corner: take away the line's y, with brackets for the negative. The corner is 2 above the line.

3. 3

   $-1-2=-3$ — Go the same distance again, below the line, so take 2 away from $-1$.

4. 4

   $(2,1)\to (2,-3)$ — Write the image. The x stays 2.

5. 5

   $(4,1)\to (4,-3)$ — Next corner: it has the same y, so the distance and the new y are the same. The x stays 4.

6. 6

   $2-(-1)=3$ — Last corner (2, 2): its distance from the line is 3.

7. 7

   $-1-3=-4,(2,2)\to (2,-4)$ — Go 3 below the line, so the new y is $-4$. The x stays 2.

Worked example

1. 1

   $y=x+1$ — The line $y=x+1$ is the line $y=x$ moved up 1. So lower the point 1 and the line becomes $y=x$.

2. 2

   $(2,5)\to (2,4)$ — Lower the point by 1: take 1 from the y.

3. 3

   $(2,4)\to (4,2)$ — Now reflect in $y=x$: swap the two numbers.

4. 4

   $(4,2)\to (4,3)$ — Raise the point by 1 to put the line back: add 1 to the y.

5. 5

   $\left(\frac{2+4}{2},\frac{5+3}{2}\right)=(3,4)$ — Check: the point halfway between (2, 5) and (4, 3) is (3, 4). Its y is its x plus 1, so it is on the mirror line.