Use the function rules to find the coordinates of A " В" C" and predict the quadrant it will be in after the transformations. ДАВС, with coordinates A (-8, 4), B (-5, 8), and C (-3, 3), is transformed

Question

b (х, у) → (x+4,4-6) to
produce AA'B'C". AA'B'C" is transformed by the
rule (x, y) → (x+ 5,1-3). Create the rule for the composition transformation and predict what quadrant the image will be in. (4 points)

Answer 1

To solve this problem, we first need to apply the given transformations to the original coordinates of points A, B, and C.

Step 1: Apply the first transformation \( (x, y) \rightarrow (x + 4, 4 - y) \).

For point A (-8, 4):

• \( A' = (-8 + 4, 4 - 4) = (-4, 0) \)

For point B (-5, 8):

• \( B' = (-5 + 4, 4 - 8) = (-1, -4) \)

For point C (-3, 3):

• \( C' = (-3 + 4, 4 - 3) = (1, 1) \)

Now the new coordinates after the first transformation are:

• \( A'(-4, 0) \)
• \( B'(-1, -4) \)
• \( C'(1, 1) \)

Step 2: Apply the second transformation \( (x, y) \rightarrow (x + 5, 1 - y) \).

For point A' (-4, 0):

• \( A'' = (-4 + 5, 1 - 0) = (1, 1) \)

For point B' (-1, -4):

• \( B'' = (-1 + 5, 1 - (-4)) = (4, 5) \)

For point C' (1, 1):

• \( C'' = (1 + 5, 1 - 1) = (6, 0) \)

New coordinates after the second transformation:

• \( A''(1, 1) \)
• \( B''(4, 5) \)
• \( C''(6, 0) \)

Step 3: Determine the quadrant of the transformed points

• Point A''(1, 1) is in Quadrant I (both x and y are positive).
• Point B''(4, 5) is also in Quadrant I.
• Point C''(6, 0) lies on the positive x-axis, which is technically not in any quadrant.

Conclusion

The coordinates of the transformed points \( A''(1, 1) \), \( B''(4, 5) \), and \( C''(6, 0) \) suggest that the transformed figure AA'B'C'' will primarily be in Quadrant I, with point C'' on the positive x-axis.

Composition Transformation Rule

The composition transformation combining both transformations can be derived as follows:

1. Initial transformation: \( (x, y) \to (x + 4, 4 - y) \)
2. Second transformation: \( (x, y) \to (x + 5, 1 - y) \)

Combining these two results gives:

• \( x' = (x + 4) + 5 = x + 9 \)
• \( y' = 1 - (4 - y) = y - 3 \)

Thus, the composition transformation rule is: \[ (x, y) \rightarrow (x + 9, y - 3) \]

The prediction for the quadrant of the image after the transformations is that the image will be primarily in Quadrant I, with point C'' on the positive x-axis.

Answer 2

Make this shorter

Answer 3

Sure! Here’s a concise version:

Transformations:

1. First Transformation: \( (x, y) \rightarrow (x + 4, 4 - y) \)

   • \( A(-8, 4) \rightarrow A'(-4, 0) \)
   • \( B(-5, 8) \rightarrow B'(-1, -4) \)
   • \( C(-3, 3) \rightarrow C'(1, 1) \)

2. Second Transformation: \( (x, y) \rightarrow (x + 5, 1 - y) \)

   • \( A'(-4, 0) \rightarrow A''(1, 1) \)
   • \( B'(-1, -4) \rightarrow B''(4, 5) \)
   • \( C'(1, 1) \rightarrow C''(6, 0) \)

Final Coordinates:

• \( A''(1, 1) \) in Quadrant I
• \( B''(4, 5) \) in Quadrant I
• \( C''(6, 0) \) on the positive x-axis

Composition Transformation Rule:

Combining the transformations gives: \[ (x, y) \rightarrow (x + 9, y - 3) \]

Quadrant Prediction:

The final figure will primarily be in Quadrant I, with point C'' on the positive x-axis.

Answer 4

Take away the “#” and the “*”s

Answer 5

Sure! Here’s the revised version without any special characters:

Transformations:

1. First Transformation: (x, y) → (x + 4, 4 - y)

   A(-8, 4) → A'(-4, 0)
   B(-5, 8) → B'(-1, -4)
   C(-3, 3) → C'(1, 1)

2. Second Transformation: (x, y) → (x + 5, 1 - y)

   A'(-4, 0) → A''(1, 1)
   B'(-1, -4) → B''(4, 5)
   C'(1, 1) → C''(6, 0)

Final Coordinates:

• A''(1, 1) in Quadrant I
• B''(4, 5) in Quadrant I
• C''(6, 0) on the positive x-axis

Composition Transformation Rule: Combining the transformations gives: (x, y) → (x + 9, y - 3)

Quadrant Prediction: The final figure will primarily be in Quadrant I, with point C'' on the positive x-axis.