Use the function rules to find the coordinates of triangle A" B" C" and predict the quadrant it will be in after the transformations. Triangle ABC, with coordinates A (-8,4) B (5,8) and C (-3,3), is transformed by (x,y)-(x+4,y-6) to produce Triangle A'B'C'. Triangle A'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.

Question

And please make your answer as simple and easy to understand as possible

Answer 1

To find the coordinates of triangle A" B" C" after the transformations, we can follow the steps below:

Step 1: Transform Triangle ABC to A'B'C'

The transformation rule is given as (x,y) → (x + 4, y - 6).

Apply to A (-8, 4):

• A' = (-8 + 4, 4 - 6) = (-4, -2)

Apply to B (5, 8):

• B' = (5 + 4, 8 - 6) = (9, 2)

Apply to C (-3, 3):

• C' = (-3 + 4, 3 - 6) = (1, -3)

The new coordinates after the first transformation are:

• A' (-4, -2)
• B' (9, 2)
• C' (1, -3)

Step 2: Transform Triangle A'B'C' to A''B''C''

The next transformation rule is (x, y) → (x + 5, y - 3).

Apply to A' (-4, -2):

• A'' = (-4 + 5, -2 - 3) = (1, -5)

Apply to B' (9, 2):

• B'' = (9 + 5, 2 - 3) = (14, -1)

Apply to C' (1, -3):

• C'' = (1 + 5, -3 - 3) = (6, -6)

The new coordinates after the second transformation are:

• A'' (1, -5)
• B'' (14, -1)
• C'' (6, -6)

Step 3: Determine the Quadrant

The coordinates A''(1, -5), B''(14, -1), and C''(6, -6) reveal that all points have a positive x-coordinate and a negative y-coordinate.

• A'' (1, -5) is in Quadrant IV.
• B'' (14, -1) is also in Quadrant IV.
• C'' (6, -6) is likewise in Quadrant IV.

Final Answer:

• The coordinates of triangle A''B''C'' are:

  • A'' (1, -5)
  • B'' (14, -1)
  • C'' (6, -6)

• Prediction: Triangle A''B''C'' will be located in Quadrant IV, where the x-coordinates are positive and the y-coordinates are negative.

Answer 2

what is the rule for the composition transformation?

Answer 3

To find the rule for the composition transformation, we need to combine the two given transformations step by step.

Given Transformations:

1. First Transformation: \[(x, y) \rightarrow (x + 4, y - 6)\]

2. Second Transformation: \[(x, y) \rightarrow (x + 5, y - 3)\]

Step-by-Step Composition:

1. Start with an initial point \((x, y)\).

2. Apply the first transformation:

   • New coordinates after the first transformation: \[(x', y') = (x + 4, y - 6)\]

3. Apply the second transformation to \((x', y')\):

   • Substituting \(x'\) and \(y'\) into the second transformation gives: \[ (x'', y'') = ((x + 4) + 5, (y - 6) - 3) \] Simplifying: \[ x'' = x + 4 + 5 = x + 9 \] \[ y'' = y - 6 - 3 = y - 9 \]

Final Composition Rule:

The composition transformation can be represented as: \[ (x, y) \rightarrow (x + 9, y - 9) \]

Summary:

The rule for the composition of the two transformations is: \[ (x, y) \rightarrow (x + 9, y - 9) \]