Use the function rules to find the coordinates of ΔA"B"C" and predict the quadrant it will be in after the transformations. ΔABC, with coordinates A(−8,4), B(−5,8), and C(−3,3), is transformed by (x,y)→(x+4,y−6) to produce ΔA′B′C′. ΔA′B′C′ is transformed by the rule (x,y)→(x+5,y−3). Create the rule for the composition transformation and predict what quadrant the image will be in.(4 points)

Answer 1

To find the coordinates of ΔA"B"C" after the transformations, we need to apply both transformation rules step by step.

1. Initial Coordinates:

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

2. First Transformation: The first transformation is given by the rule \((x, y) \to (x + 4, y - 6)\).

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

   So, the coordinates after the first transformation are:

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

3. Second Transformation: The second transformation is given by the rule \((x, y) \to (x + 5, y - 3)\).

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

   So, the coordinates after the second transformation are:

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

4. Final Coordinates: After both transformations, the coordinates of \( \Delta A''B''C'' \) are:

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

5. Quadrant Prediction:

• The coordinates \( (1, -5) \), \( (4, -1) \), and \( (6, -6) \) all have positive x-coordinates and negative y-coordinates.
• This means that the points are located in the Fourth Quadrant.

Summary:

The final coordinates of \( \Delta A''B''C'' \) are:

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

And the shape will be in the Fourth Quadrant.