Question

consider triangle abc graphed here.
here are some transformation rules. for each transformation, first predict what the image of triangle abc will look like. then compute the coordinates of the image and draw it.
a. ((x,y) \to (x - 4, y - 1)
type your answers in the boxes.
(a): (\boxed{}) (\boxed{})
(b): (\boxed{}) (\boxed{})
(c): (\boxed{}) (\boxed{})
b. ((x,y) \to (y, x)
type your answers in the boxes.
(a): (\boxed{}) (\boxed{})

Explanation:

Step1: Identify Original Coordinates

First, find the coordinates of \( A \), \( B \), \( C \) from the graph.

• \( A \) is at \( (2, 3) \)
• \( B \) is at \( (4, 2) \)
• \( C \) is at \( (3, 4) \)

Step2: Apply Transformation (a) \((x,y) \to (x - 4, y - 1)\)

For \( A' \):
Substitute \( x = 2 \), \( y = 3 \) into \( (x - 4, y - 1) \):
\( 2 - 4 = -2 \), \( 3 - 1 = 2 \) → \( A' = (-2, 2) \)

For \( B' \):
Substitute \( x = 4 \), \( y = 2 \) into \( (x - 4, y - 1) \):
\( 4 - 4 = 0 \), \( 2 - 1 = 1 \) → \( B' = (0, 1) \)

For \( C' \):
Substitute \( x = 3 \), \( y = 4 \) into \( (x - 4, y - 1) \):
\( 3 - 4 = -1 \), \( 4 - 1 = 3 \) → \( C' = (-1, 3) \)

Step3: Apply Transformation (b) \((x,y) \to (y, x)\)

For \( A' \):
Substitute \( x = 2 \), \( y = 3 \) into \( (y, x) \):
\( (3, 2) \) → \( A' = (3, 2) \)

Part (a) Answers:

\( A' : \boldsymbol{(-2, 2)} \)
\( B' : \boldsymbol{(0, 1)} \)
\( C' : \boldsymbol{(-1, 3)} \)

Part (b) Answer (for \( A' \)):

\( A' : \boldsymbol{(3, 2)} \)

Answer:

Step1: Identify Original Coordinates

First, find the coordinates of \( A \), \( B \), \( C \) from the graph.

• \( A \) is at \( (2, 3) \)
• \( B \) is at \( (4, 2) \)
• \( C \) is at \( (3, 4) \)

Step2: Apply Transformation (a) \((x,y) \to (x - 4, y - 1)\)

For \( A' \):
Substitute \( x = 2 \), \( y = 3 \) into \( (x - 4, y - 1) \):
\( 2 - 4 = -2 \), \( 3 - 1 = 2 \) → \( A' = (-2, 2) \)

For \( B' \):
Substitute \( x = 4 \), \( y = 2 \) into \( (x - 4, y - 1) \):
\( 4 - 4 = 0 \), \( 2 - 1 = 1 \) → \( B' = (0, 1) \)

For \( C' \):
Substitute \( x = 3 \), \( y = 4 \) into \( (x - 4, y - 1) \):
\( 3 - 4 = -1 \), \( 4 - 1 = 3 \) → \( C' = (-1, 3) \)

Step3: Apply Transformation (b) \((x,y) \to (y, x)\)

For \( A' \):
Substitute \( x = 2 \), \( y = 3 \) into \( (y, x) \):
\( (3, 2) \) → \( A' = (3, 2) \)

Part (a) Answers:

\( A' : \boldsymbol{(-2, 2)} \)
\( B' : \boldsymbol{(0, 1)} \)
\( C' : \boldsymbol{(-1, 3)} \)

Part (b) Answer (for \( A' \)):

\( A' : \boldsymbol{(3, 2)} \)