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use the function rules to find the coordinates of A" B" C" and predict the quadrant it will be in after the transformations. ABC, with coordinates A(-8,4), b(-5,8), and C(-3,3), is transformed by (x,y) > (x + 4, y - 6) to produce A'B'C'. A'B'C' is transformed by the rule (x,y) > (x + 5, y - 3). Create the rule fore the composition transformation and predict what quadrant the image will be in. Please simplify explanation

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Let's break down the transformations step-by-step to find the coordinates of points A", B", and C" and determine the quadrant in which the final image will be.

Step 1: First Transformation

The first transformation rule is: \[ (x, y) \to (x + 4, y - 6) \] We will apply this to each point A, B, and C.

  • Point A(-8, 4): \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]

  • Point B(-5, 8): \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]

  • Point C(-3, 3): \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]

Now the coordinates of A', B', and C' are:

  • A'(-4, -2)
  • B'(-1, 2)
  • C'(1, -3)

Step 2: Second Transformation

The second transformation rule is: \[ (x, y) \to (x + 5, y - 3) \] We will apply this to points A', B', and C'.

  • Point A'(-4, -2): \[ A'' = (-4 + 5, -2 - 3) = (1, -5) \]

  • Point B'(-1, 2): \[ B'' = (-1 + 5, 2 - 3) = (4, -1) \]

  • Point C'(1, -3): \[ C'' = (1 + 5, -3 - 3) = (6, -6) \]

Now the coordinates of A'', B'', and C'' are:

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

Step 3: Composition Transformation Rule

To find a single rule for the composition of both transformations, we can combine them:

  1. From the first transformation: \((x, y) \to (x + 4, y - 6)\)
  2. Then apply the second transformation to the result: \((x + 4, y - 6) \to (x + 4 + 5, y - 6 - 3)\)

This simplifies to: \[ (x, y) \to (x + 9, y - 9) \]

Step 4: Predict the Quadrant

Now, let's analyze where the final coordinates A'', B'', and C'' are located:

  • Point A''(1, -5) is located in Quadrant IV (positive x, negative y).
  • Point B''(4, -1) is also in Quadrant IV (positive x, negative y).
  • Point C''(6, -6) is also in Quadrant IV (positive x, negative y).

Since all three points are in Quadrant IV, we can conclude that the final image of A'', B'', and C'' will be in Quadrant IV.