Use the function rules to find the coordinates of A"B" C"𝐴" 𝐵"𝐶" and predict the quadrant it will be in after the transformations. Draw a picture to help.△ABC△𝐴𝐵𝐶 coordinates: A(−8,4)𝐴(−8,4), B(−5,8)𝐵(−5,8), C(−3,3)𝐶(−3,3)△ABC△𝐴𝐵𝐶 to △A′B′C′△𝐴′𝐵′𝐶′ rule: (x,y)→(x+4, y−6)(𝑥,𝑦)→(𝑥+4, 𝑦−6)△A′B′C′△𝐴′𝐵′𝐶′ to △A"B" C"△𝐴" 𝐵"𝐶" rule: (x,y)→(x+5, y−3)(𝑥,𝑦)→(𝑥+5, 𝑦−3)Show your work for each transformation. Create the rule for the composition transformation. Predict what quadrant the image will be in.(4 points)

Answer 1

To find the coordinates of points \( A'', B'', C'' \) after applying the transformations as specified, we'll follow these steps:

1. Initial Coordinates of Triangle \(\triangle ABC\):

   • \( A(-8, 4) \)
   • \( B(-5, 8) \)
   • \( C(-3, 3) \)

2. First Transformation Rule: \((x,y) \rightarrow (x+4, y-6)\):

   • For point \( A \): \[ A' = A(-8, 4) \rightarrow (-8 + 4, 4 - 6) = (-4, -2) \]
   • For point \( B \): \[ B' = B(-5, 8) \rightarrow (-5 + 4, 8 - 6) = (-1, 2) \]
   • For point \( C \): \[ C' = C(-3, 3) \rightarrow (-3 + 4, 3 - 6) = (1, -3) \]

   After the first transformation, the new coordinates are: \[ A'(-4, -2), \quad B'(-1, 2), \quad C'(1, -3) \]

3. Second Transformation Rule: \((x,y) \rightarrow (x+5, y-3)\):

   • For point \( A' \): \[ A'' = A'(-4, -2) \rightarrow (-4 + 5, -2 - 3) = (1, -5) \]
   • For point \( B' \): \[ B'' = B'(-1, 2) \rightarrow (-1 + 5, 2 - 3) = (4, -1) \]
   • For point \( C' \): \[ C'' = C'(1, -3) \rightarrow (1 + 5, -3 - 3) = (6, -6) \]

   After the second transformation, the new coordinates are: \[ A''(1, -5), \quad B''(4, -1), \quad C''(6, -6) \]

4. Composition Rule: The entire transformation can be represented as a single combined rule.

   The first transformation is represented by: \[ (x, y) \rightarrow (x + 4, y - 6) \] The second transformation takes the output from the first: \[ (x + 4, y - 6) \rightarrow ((x + 4) + 5, (y - 6) - 3) \] which simplifies to: \[ (x, y) \rightarrow (x + 9, y - 9) \]

5. Predicting the Quadrant: The coordinates of \( A'', B'', C'' \) are:

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

   A point in the coordinate plane is in quadrant IV if \( x > 0 \) and \( y < 0 \).

   Since all the transformed points have positive \( x \)-coordinates and negative \( y \)-coordinates, it follows that: \[ A'', B'', C'' \text{ are all in Quadrant IV.} \]

6. Sketching the Transformed Triangle: Unfortunately, I can't draw directly here, but you would simply plot points \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \) to visualize that they lie in quadrant IV.

Final notation of coordinates after transformations: \[ A''(1, -5), \quad B''(4, -1), \quad C''(6, -6) \]

Final Prediction: The image \( \triangle A''B''C'' \) will be located in Quadrant IV.