Transformations in the Coordinate Plane: Translations, Reflections, and Rotations

Section 2.1: Translations in the Coordinate Plane

• Overview of Translations:

  • A translation is a transformation that slides a figure from one position to another without changing its size or shape.

• Example 2: Translating a Figure in the Coordinate Plane:

  • Problem Statement: Translate the red triangle $3$ units right and $3$ units down. Find the coordinates of the image.
  • Original Vertices:
  • $A(-2, 1)$
  • $B(2, 5)$
  • $C(1, 2)$

• Method 1: Graphical Approach (Using a Coordinate Plane):

  1. Move each vertex of triangle $ABC$ $3$ units to the right and $3$ units down on the grid.
  2. Connect the translated vertices to form the image.
  3. Label the new vertices as $A'$, $B'$, and $C'$.
  4. Read the coordinates directly from the coordinate plane:
     • $A'(1, -2)$
     • $B'(5, 2)$
     • $C'(4, -1)$

• Method 2: Algebraic Approach (Using Coordinate Rules):

  1. Add $3$ to the x-coordinates of each original vertex to shift the figure $3$ units to the right.
  2. Add $-3$ (or subtract $3$) from the y-coordinates of each original vertex to shift the figure $3$ units down.
  3. Perform calculations for each vertex:
     • Vertex $A$:          $A(-2, 1) \rightarrow A'(-2 + 3, 1 - 3) \rightarrow A'(1, -2)$
     • Vertex $B$:          $B(2, 5) \rightarrow B'(2 + 3, 5 - 3) \rightarrow B'(5, 2)$
     • Vertex $C$:          $C(1, 2) \rightarrow C'(1 + 3, 2 - 3) \rightarrow C'(4, -1)$
  4. The coordinates of the resulting image are $A'(1, -2)$, $B'(5, 2)$, and $C'(4, -1)$.
  • Try It Exercises:

• Exercise 3 (WHAT IF?):

  • Problem: The red triangle with original vertices $A(-2, 1)$, $B(2, 5)$, and $C(1, 2)$ is translated $4$ units left and $2$ units up. What are the coordinates of the image?

  • Algebraic Translation Procedure:

    • Subtract $4$ from the x-coordinates and add $2$ to the y-coordinates:         $(x, y) \rightarrow (x - 4, y + 2)$
    • $A(-2, 1) \rightarrow A'(-2 - 4, 1 + 2) \rightarrow A'(-6, 3)$
    • $B(2, 5) \rightarrow B'(2 - 4, 5 + 2) \rightarrow B'(-2, 7)$
    • $C(1, 2) \rightarrow C'(1 - 4, 2 + 2) \rightarrow C'(-3, 4)$

  • Image Coordinates: $A'(-6, 3)$, $B'(-2, 7)$, and $C'(-3, 4)$.

  • Self-Assessment for Concepts & Skills:

• Identifying a Translation:

  • Exercise 4: Compare the red figure and blue figure. Determine if the blue figure is a translation of the red figure.
    • Explanation: The blue parallelogram has the same orientation and shape as the red parallelogram, shifted diagonally. Thus, it is a translation.
  • Exercise 5: Compare the red triangle and blue triangle. Determine if the blue figure is a translation of the red figure.
    • Explanation: The red triangle points downward while the blue triangle points upward (inverted). Because the orientation has changed, the blue figure is not a translation of the red figure.

• Translating a Figure Exercise:

  • Exercise 6:
    • Given: Vertices of a triangle are $A(-2, -2)$, $B(0, 2)$, and $C(3, 0)$.
    • Transformation: Translate the triangle $1$ unit left and $2$ units up.
    • Coordinate Rule: $(x, y) \rightarrow (x - 1, y + 2)$
    • Step-by-Step Calculation:
    • $A(-2, -2) \rightarrow A'(-2 - 1, -2 + 2) \rightarrow A'(-3, 0)$
    • $B(0, 2) \rightarrow B'(0 - 1, 2 + 2) \rightarrow B'(-1, 4)$
    • $C(3, 0) \rightarrow C'(3 - 1, 0 + 2) \rightarrow C'(2, 2)$
    • Image Coordinates: $A'(-3, 0)$, $B'(-1, 4)$, and $C'(2, 2)$.

Section 2.2: Reflections in the Coordinate Plane

• Overview of Reflections:

  • A reflection is a transformation that flips a figure over a line of reflection (such as the x-axis or y-axis) to create a mirror image.

• Example 2: Reflecting Figures:

  • Given Vertices: Triangle with vertices $A(1, 1)$, $B(1, 4)$, and $C(3, 4)$.
  • Goal: Draw the figure and its reflection in (a) the x-axis and (b) the y-axis, and state the coordinates of the image.

• Part (a): Reflection in the x-axis:

  • Geometric Method:
    • Point $A$ is $1$ unit above the x-axis; plot image point $A'$ $1$ unit below the x-axis.
    • Points $B$ and $C$ are $4$ units above the x-axis; plot image points $B'$ and $C'$ $4$ units below the x-axis.
    • Coordinates of image: $A'(1, -1)$, $B'(1, -4)$, and $C'(3, -4)$.
  • Coordinate Rule (Another Method):
    • Take the opposite of each y-coordinate while keeping x-coordinates unchanged:         $(x, y) \rightarrow (x, -y)$
    • $A(1, 1) \rightarrow A'(1, -1)$
    • $B(1, 4) \rightarrow B'(1, -4)$
    • $C(3, 4) \rightarrow C'(3, -4)$

• Part (b): Reflection in the y-axis:

  • Geometric Method:

    • Points $A$ and $B$ are $1$ unit to the right of the y-axis; plot image points $A'$ and $B'$ $1$ unit to the left of the y-axis.
    • Point $C$ is $3$ units to the right of the y-axis; plot image point $C'$ $3$ units to the left of the y-axis.
    • Coordinates of image: $A'(-1, 1)$, $B'(-1, 4)$, and $C'(-3, 4)$.

  • Coordinate Rule:

    • Take the opposite of each x-coordinate while keeping y-coordinates unchanged:         $(x, y) \rightarrow (-x, y)$
    • $A(1, 1) \rightarrow A'(-1, 1)$
    • $B(1, 4) \rightarrow B'(-1, 4)$
    • $C(3, 4) \rightarrow C'(-3, 4)$

  • Try It Exercises:

• Exercise 3:

  • Given: Rectangle vertices $A(-4, -3)$, $B(-4, -1)$, $C(-1, -1)$, and $D(-1, -3)$.

  • Part (a) Reflection in the x-axis:

    • Rule: $(x, y) \rightarrow (x, -y)$
    • $A(-4, -3) \rightarrow A'(-4, 3)$
    • $B(-4, -1) \rightarrow B'(-4, 1)$
    • $C(-1, -1) \rightarrow C'(-1, 1)$
    • $D(-1, -3) \rightarrow D'(-1, 3)$

  • Part (b) Reflection in the y-axis:

    • Rule: $(x, y) \rightarrow (-x, y)$
    • $A(-4, -3) \rightarrow A'(4, -3)$
    • $B(-4, -1) \rightarrow B'(4, -1)$
    • $C(-1, -1) \rightarrow C'(1, -1)$
    • $D(-1, -3) \rightarrow D'(1, -3)$

  • Self-Assessment for Concepts & Skills:

• Exercise 4: Reflecting a Figure:

  • Given: Vertices $J(-3, -5)$, $K(-2, 2)$, and $L(1, -4)$.
  • Part (a) Reflection in the x-axis:
    • $J(-3, -5) \rightarrow J'(-3, 5)$
    • $K(-2, 2) \rightarrow K'(-2, -2)$
    • $L(1, -4) \rightarrow L'(1, 4)$
  • Part (b) Reflection in the y-axis:
    • $J(-3, -5) \rightarrow J'(3, -5)$
    • $K(-2, 2) \rightarrow K'(2, 2)$
    • $L(1, -4) \rightarrow L'(-1, -4)$

• Exercise 5: Which One Doesn't Belong?:

  • Task: Determine which transformation of a baseball cap figure does not belong with the other three and explain reasoning.
  • Analysis:
    • Option 1: Two hats facing opposite directions mirrored across a vertical line (Reflection).
    • Option 2: Two hats facing the exact same direction shifted horizontally (Translation).
    • Option 3: Two hats facing opposite directions mirrored across a vertical line (Reflection).
    • Option 4: Two hats facing opposite directions mirrored across a vertical line (Reflection).
  • Conclusion & Reasoning: The second pair does not belong because it represents a translation (slide) where the orientation remains identical. The other three pairs represent reflections across a vertical line.

Section 2.3: Rotations in the Coordinate Plane

• Key Concepts & Rules for Rotations:
  • Equivalence of Rotation Angles:
  • A counterclockwise rotation of $n^\circ$ is equivalent to a clockwise rotation of $(360 - n)^\circ$.
  • A clockwise rotation of $n^\circ$ is equivalent to a counterclockwise rotation of $(360 - n)^\circ$.
  • Congruence / Identity:
  • In a rotation, the original figure and its image are identical (congruent).
  • Coordinate Rules for Counterclockwise Rotations About the Origin $(0, 0)$:
  • $90^\circ$ Counterclockwise Rotation:       $(x, y) \rightarrow (-y, x)$
  • $180^\circ$ Rotation:       $(x, y) \rightarrow (-x, -y)$
  • $270^\circ$ Counterclockwise Rotation:       $(x, y) \rightarrow (y, -x)$
  • Direction Directionality for $180^\circ$ Rotations:
  • A $180^\circ$ clockwise rotation and a $180^\circ$ counterclockwise rotation produce the exact same image. Therefore, direction does not need to be specified when rotating a figure $180^\circ$.

• Example 2: Rotating a Figure:

  • Problem: Vertices of a trapezoid are $W(-4, 2)$, $X(-3, 4)$, $Y(-1, 4)$, and $Z(-1, 2)$. Rotate the trapezoid $180^\circ$ about the origin. What are the coordinates of the image?
  • Applied Coordinate Rule:
  • A point $(x, y)$ rotated $180^\circ$ about the origin results in an image with coordinates $(-x, -y)$.
  • Step-by-Step Coordinate Transformations:
  • $W(-4, 2) \rightarrow W'(4, -2)$
  • $X(-3, 4) \rightarrow X'(3, -4)$
  • $Y(-1, 4) \rightarrow Y'(1, -4)$
  • $Z(-1, 2) \rightarrow Z'(1, -2)$
  • Image Coordinates: $W'(4, -2)$, $X'(3, -4)$, $Y'(1, -4)$, and $Z'(1, -2)$.

• Try It Exercises:

  • Exercise 3:
  • Given Vertices: $J(-4, -4)$, $K(-4, 2)$, $L(-1, 0)$, and $M(-2, -3)$.
  • Transformation: Rotate $180^\circ$ about the origin.
  • Rule: $(x, y) \rightarrow (-x, -y)$
  • Calculations:
    • $J(-4, -4) \rightarrow J'(4, 4)$
    • $K(-4, 2) \rightarrow K'(4, -2)$
    • $L(-1, 0) \rightarrow L'(1, 0)$
    • $M(-2, -3) \rightarrow M'(2, 3)$
  • Image Coordinates: $J'(4, 4)$, $K'(4, -2)$, $L'(1, 0)$, and $M'(2, 3)$.
  • Exercise 4:
  • Given Vertices: $P(-3, 2)$, $Q(6, 1)$, and $R(-1, -5)$.
  • Transformation: Rotate $90^\circ$ counterclockwise about the origin.
  • Rule: $(x, y) \rightarrow (-y, x)$
  • Calculations:
    • $P(-3, 2) \rightarrow P'(-2, -3)$
    • $Q(6, 1) \rightarrow Q'(-1, 6)$
    • $R(-1, -5) \rightarrow R'(5, -1)$
  • Image Coordinates: $P'(-2, -3)$, $Q'(-1, 6)$, and $R'(5, -1)$.
  • Exercise 5:
  • Given Vertices: $A(5, 3)$, $B(4, -1)$, and $C(1, -1)$.
  • Transformation: Rotate $90^\circ$ clockwise about the origin.
  • Equivalence Note: A $90^\circ$ clockwise rotation is equivalent to a $270^\circ$ counterclockwise rotation.
  • Rule: $(x, y) \rightarrow (y, -x)$
  • Calculations:
    • $A(5, 3) \rightarrow A'(3, -5)$
    • $B(4, -1) \rightarrow B'(-1, -4)$
    • $C(1, -1) \rightarrow C'(-1, -1)$
  • Image Coordinates: $A'(3, -5)$, $B'(-1, -4)$, and $C'(-1, -1)$.