4.4 Describing the impact of a transformation matrix on a graphical object

• A rotation matrix rotates objects counterclockwise around the origin.

  True
• A rotation matrix uses the angle \theta to rotate objects around the origin.
• What does a dilation matrix do to an object?

  Scales the object
• $\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ are scaling factors
• What are transformation matrices used for in a coordinate plane?

  Geometric transformations
• A scaling matrix changes the size of an object by multiplying its coordinates by factors $a$ and b</latex>.


  True
• Which matrix reflects an object across the x-axis?

  \begin{bmatrix} 1 & 0 \\ 0 & - 1 \end{bmatrix}</latex>
• A reflection matrix across the y-axis is represented by \begin{bmatrix} - 1 & 0 \\ 0 & 1 \end{bmatrix}</latex>.


  True
• A shear matrix uses the factor k to shift points.
• A shear matrix shifts points in one direction based on their position in another direction.


  True
• The identity matrix leaves objects unchanged.

  True
• A reflection matrix reflects objects across an axis.


  True
• Steps to apply a geometric transformation to a graphical object:

  1️⃣ Represent the object as a matrix

  2️⃣ Choose the appropriate transformation matrix

  3️⃣ Multiply the object matrix by the transformation matrix

  4️⃣ Obtain the transformed object matrix
• Scaling, rotation, and reflection transformations can be applied by multiplying the object matrix by the corresponding transformation matrix
• What are the new coordinates of the point $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ after applying a scaling transformation with factors 2 and 3 in the x and y directions, respectively?


  (6, 6)
• What happens to an object's orientation after applying a rotation matrix?

  It rotates
• How can a rotation matrix be visualized in GeoGebra?

  By rotating a triangle
• What is the rotation matrix for a 90-degree counterclockwise rotation?

  $\begin{bmatrix} \cos \frac{\pi}{2} & - \sin \frac{\pi}{2} \\ \sin \frac{\pi}{2} & \cos \frac{\pi}{2} \end{bmatrix}$
• A translation matrix shifts the object by adding offset coordinates
• Fundamental transformation matrices include rotation, reflection, dilation, and shear.


  True
• Match the transformation matrix with its description:

  $\begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ ↔️ Rotation

  $\begin{bmatrix} 1 & 0 \\ 0 & - 1 \end{bmatrix}$ ↔️ Reflection across x-axis

  $\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ ↔️ Dilation

  $\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$ ↔️ Shear along x-axis
• What is the rotation matrix when θ = 45 degrees?

  $\begin{bmatrix} \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$
• Steps to apply a scaling transformation to a point using matrices

  1️⃣ Represent the point as a matrix

  2️⃣ Multiply the point matrix by the scaling matrix

  3️⃣ Calculate the resulting matrix
• What is the transformed coordinate of the point $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ after applying the scaling matrix $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$?


  $\begin{bmatrix} 6 \\ 6 \end{bmatrix}$
• What is the transformed coordinate of the point $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ after applying a reflection across the x-axis?


  $\begin{bmatrix} 3 \\ - 2 \end{bmatrix}$
• Geometric transformations such as scaling, rotation, and reflection can be applied by multiplying the object matrix by the corresponding transformation matrix
• What changes in an object's properties after applying a rotation transformation matrix?

  Orientation changes
• Desmos is a graphical tool that allows users to visualize scaling and rotation
• The scaling matrix $\begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}$ transforms the point $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ to the new coordinates (6, 6)
• Common geometric transformations performed by transformation matrices include scaling, rotation, and reflection
• A rotation matrix rotates objects around the origin by an angle \theta
• A shear matrix shifts points based on their position in another direction
• What is the resulting matrix after applying the scaling transformation $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ to the point matrix $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$?


  $\begin{bmatrix} 6 \\ 6 \end{bmatrix}$
• Applying a transformation matrix to an object's matrix representation changes its coordinates according to the specified transformation.


  True
• Match the graphical tool with its features:

  GeoGebra ↔️ Interactive canvas

  Desmos ↔️ Graphing interface
• A scaling matrix with factors a</latex> and $b$ scales the x and y coordinates by these factors
• A shear along the x-axis shifts points in the x direction based on their y-coordinate.

  True
• A translation moves an object by adding offset coordinates
• Reflection matrices reflect objects across the x-axis or the y-axis
• A shear transformation shifts points in one direction based on their position in another direction