Question

name
1 consider δabc.

a translate δabc 3 units right. label the image abc. how are the values in the ordered pairs affected by the translation?

b translate δabc 5 units down. label the image abc. how are the values in the ordered pairs affected by the translation?

c how could you determine the coordinates of the vertices of a translated image without using a graph?

Explanation:

Step1: Identify Original Coordinates

From the graph, find coordinates of \( A \), \( B \), \( C \).
\( A(-6, 7) \), \( B(-4, 2) \), \( C(-8, 2) \).

Step2: Translate 3 Units Right (Part a)

For a horizontal translation \( h \) units right, new \( x \)-coordinate = original \( x + h \), \( y \)-coordinate remains.

• \( A'(-6 + 3, 7) = (-3, 7) \)
• \( B'(-4 + 3, 2) = (-1, 2) \)
• \( C'(-8 + 3, 2) = (-5, 2) \)

Effect: \( x \)-values increase by 3, \( y \)-values unchanged.

Step3: Translate 5 Units Down (Part b)

For a vertical translation \( k \) units down, new \( y \)-coordinate = original \( y - k \), \( x \)-coordinate remains.

• \( A''(-6, 7 - 5) = (-6, 2) \)
• \( B''(-4, 2 - 5) = (-4, -3) \)
• \( C''(-8, 2 - 5) = (-8, -3) \)

Effect: \( y \)-values decrease by 5, \( x \)-values unchanged.

Step4: Determine Coordinates Without Graph (Part c)

Use translation rules:

• Horizontal translation: Add (right) or subtract (left) from \( x \)-coordinate.
• Vertical translation: Add (up) or subtract (down) from \( y \)-coordinate.

Apply these to original coordinates.

Answer:

Part (a)

• \( A'(-3, 7) \), \( B'(-1, 2) \), \( C'(-5, 2) \)
• Effect: \( x \)-coordinates increase by 3, \( y \)-coordinates stay the same.

Part (b)

• \( A''(-6, 2) \), \( B''(-4, -3) \), \( C''(-8, -3) \)
• Effect: \( y \)-coordinates decrease by 5, \( x \)-coordinates stay the same.

Part (c)

Use translation rules: For a translation \( (h, k) \) (right/left \( h \), up/down \( k \)), new coordinates are \( (x + h, y + k) \) (right/up: positive \( h/k \); left/down: negative \( h/k \)). Apply to original vertices’ coordinates.