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.
A

z

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1 answer

To find the coordinates of \( \Delta A"B"C" \) after the transformations, we will apply each transformation step by step.

Initial Coordinates

The original coordinates of the triangle \( \Delta ABC \) are:

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

First Transformation

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

We apply this to each vertex of \( \Delta ABC \):

  • 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 of \( \Delta A'B'C' \) are:

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

Second Transformation

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

We will now apply this transformation to each vertex of \( \Delta A'B'C' \):

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

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

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

Thus, the coordinates of \( \Delta A"B"C" \) after both transformations are:

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

Composition Rule

To create the composition transformation from the original coordinates \( (x, y) \) to \( (x + 9, y - 9) \) (adding both transformations together):

  • The total transformation can be viewed as: \[ (x, y) \to (x + 4 + 5, y - 6 - 3) \to (x + 9, y - 9) \]

Quadrant Prediction

Now, let's analyze the final coordinates \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \):

  • All these points have a positive x-coordinate and a negative y-coordinate. This means:
    • \( x > 0 \) and \( y < 0 \)

Therefore, the points lie in Quadrant IV.

Conclusion

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

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

The triangle \( \Delta A"B"C" \) will be in Quadrant IV after the transformations.