Example Vector:

Worked Example 1

Let's translate a blue object by the vector $\begin{pmatrix} 8 \\ 5 \end{pmatrix}$:

1. Start with the blue object at its initial position on the grid.
2. Move each point of the object $8$ units to the right and $5$ units up.
3. Draw the new object in green at the new position.

Worked Example 2

Translate the blue object by the vector $\begin{pmatrix} 5 \\ -5 \end{pmatrix}$:

4. Start with the blue object on the grid.
5. Move each point of the object $5$ units to the right and $5$ units down (note the negative value).
6. Draw the new object in green at the new position.

Worked Example 1: Reflection in a Vertical Line

Reflect the blue object in the red line, given by the equation $x=2$:

7. Identify the line of reflection: The line $x=2$ is a vertical line passing through the grid where $x=2$.
8. Reflect each point of the blue object across the line $x=2$.
9. Draw the reflected object (purple) on the other side of the line, ensuring each point is the same distance from the line as the corresponding original point. Notice: Every point on the purple object is the same distance from the line of reflection as the corresponding point on the blue object.

Worked Example 2: Reflection in a Diagonal Line

Reflect the blue object in the red line, given by the equation $y=x$:

10. Identify the line of reflection: The line $y=x$ is a diagonal line where every $x$ value equals its corresponding $y$ value.
11. Reflect each point of the blue object across the line $y=x$.
12. Draw the reflected object (purple) on the other side of the line, ensuring each point is equidistant from the line. Notice: Reflecting across a diagonal line like $y=x$ can be more challenging, but the principle remains the same: each point on the reflected object is the same distance from the mirror line as the corresponding original point.

Worked Example 1: Rotating Around the Origin ($0, 0$)

Rotate the blue object to obtain the purple object with the following details:

• Centre of Rotation: ($0, 0$) - the origin
• Direction of Rotation: Clockwise
• Angle of Rotation: $90°$ Steps:

13. Locate the centre of rotation at ($0, 0$) on the grid.
14. Rotate each point of the blue object $90°$ clockwise around the origin.
15. Plot the new points and draw the purple object. Notice: If you wanted to describe the same rotation differently, you could say it was a $270°$ anti-clockwise rotation since both produce the same result.

Worked Example 2: Rotating Around a Point ($2, 1$)

Rotate the blue object to obtain the purple object with the following details:

• Centre of Rotation: ($2, 1$)
• Direction of Rotation: Clockwise
• Angle of Rotation: $180°$ Steps:

16. Locate the centre of rotation at ($2, 1$) on the grid.
17. Rotate each point of the blue object $180°$ clockwise around ($2, 1$).
18. Plot the new points and draw the purple object. Notice: Whenever the angle of rotation is $180°$, the direction (clockwise or anti-clockwise) does not matter—the result is the same.

Examples:

• If the scale factor is $3$, the new shape will be three times as large.
• If the scale factor is $1/2$, the new shape will be half the size of the original

Worked Example 1: Enlarging a Shape

Given:

• Centre of Enlargement: ($-8, -6$)
• Scale Factor: $2$ Steps:

19. Draw lines from the centre of enlargement through each vertex (corner) of the blue object.
20. Measure the distance from the centre of enlargement to each vertex of the blue shape.
21. Double the distance to find where the vertices of the purple shape should be, as the scale factor is $2$.
22. Plot the new points and draw the enlarged purple shape. Notice: The distance of each point in the purple shape from the centre of enlargement is twice that of the corresponding point on the blue shape.

Worked Example 2: Reducing a Shape (Fractional Scale Factor)

Given:

• Centre of Enlargement: ($-6, 5$)
• Scale Factor: $1/3$ Steps:

23. Draw lines from the centre of enlargement through each vertex of the blue object.
24. Measure the distance from the centre of enlargement to each vertex of the blue shape.
25. Reduce the distance by a factor of $1/3$ to find where the vertices of the purple shape should be.
26. Plot the new points and draw the reduced purple shape. Notice: Since the shape got smaller, the scale factor is fractional (less than $1$). Each point on the purple shape is one-third the distance from the centre of enlargement as the corresponding point on the blue shape.