Use the function rules to find the coordinates of%C2%A0ΔA"B"C" %C2%A0and predict the quadrant it will be in after the transformations.%C2%A0ΔABC ,%C2%A0with coordinates%C2%A0A(−8,4) ,%C2%A0B(−5,8) , and%C2%A0C(−3,3) ,%C2%A0is transformed by%C2%A0(x,y)→(x+4,y−6) %C2%A0to produce%C2%A0ΔA′B′C′ .%C2%A0ΔA′B′C′ %C2%A0is transformed by the rule%C2%A0(x,y)→(x+5,y−3) .%C2%A0Create the rule for the composition transformation and predict what quadrant the image will be in.(4 points

Answer 1

To find the coordinates of \(\Delta A''B''C''\) after performing the transformations on \(\Delta ABC\) with the given coordinates of points, let's follow the transformations step-by-step.

Step 1: Transform \(\Delta ABC\) to \(\Delta A'B'C'\)

Original Points:

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

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

Applying the transformation:

1. For point \(A\): \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]

2. For point \(B\): \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]

3. For point \(C\): \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]

So, after the first transformation, the new coordinates are:

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

Step 2: Transform \(\Delta A'B'C'\) to \(\Delta A''B''C''\)

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

Applying the transformation:

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

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

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

So, after the second transformation, the new coordinates are:

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

Summary of Final Coordinates

The final coordinates after both transformations are:

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

Step 3: Predict the Quadrant

The coordinates of the points are:

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

Since all the transformed points \(A'', B'', C''\) have positive \(x\) coordinates and negative \(y\) coordinates, they all lie in Quadrant IV.

Composition Rule and Conclusion

The overall composition transformation from \(\Delta ABC\) to \(\Delta A''B''C''\) can be represented as:

\[ (x, y) \rightarrow (x + 4, y - 6) \rightarrow (x + 4 + 5, y - 6 - 3) \rightarrow (x + 9, y - 9) \]

So, the composition rule is: \[ (x, y) \rightarrow (x + 9, y - 9) \]

Based on the calculations, we predict that the image will be in Quadrant IV after the transformations.

Answer 2

in a essay

Answer 3

Transformations of Triangle \(\Delta ABC\): A Detailed Analysis

The study of transformations in geometry is a fundamental concept that helps us understand the relationship between shapes and their positions in a coordinate plane. In this essay, we will explore a series of transformations applied to triangle \(\Delta ABC\) with the coordinates \(A(-8, 4)\), \(B(-5, 8)\), and \(C(-3, 3)\). Through careful application of transformation rules, we will determine the final coordinates of the triangle after two successive transformations and predict in which quadrant the transformed triangle will lie.

Original Coordinates of Triangle \(\Delta ABC\)

Initially, we start with the coordinates of the vertices of triangle \(\Delta ABC\):

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

These coordinates place the triangle predominantly in Quadrant II, where \(x\) values are negative and \(y\) values are positive.

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

The first transformation applied to the triangle shifts each point right by 4 units and down by 6 units. This can be represented by the rule \((x, y) \rightarrow (x + 4, y - 6)\).

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

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

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

After this transformation, the new coordinates for the vertices of \(\Delta A'B'C'\) are:

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

Upon examining these new coordinates, we note that point \(A'\) and point \(C'\) now occupy Quadrant III because \(A'(-4, -2)\) has a negative \(x\) and a negative \(y\) value, while point \(B'(-1, 2)\) remains in Quadrant II since it still has a negative \(x\) and a positive \(y\).

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

The second transformation again shifts each point, this time to the right by 5 units and down by 3 units, described by the rule \((x, y) \rightarrow (x + 5, y - 3)\).

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

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

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

After the second transformation, the coordinates of the vertices of \(\Delta A''B''C''\) are:

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

Final Coordinates and Quadrant Prediction

Upon examining the final coordinates, we observe that:

• Point \(A''(1, -5)\) has a positive \(x\) and a negative \(y\).
• Point \(B''(4, -1)\) also has a positive \(x\) and a negative \(y\).
• Point \(C''(6, -6)\) follows the same trend.

Given that all transformed points now lie where the \(x\) coordinates are positive and the \(y\) coordinates are negative, it is evident that triangle \(\Delta A''B''C''\) resides entirely in Quadrant IV.

Conclusion: The Composition of Transformations

In conclusion, the series of transformations applied to triangle \(\Delta ABC\) not only significantly altered the coordinates of its vertices but also shifted it from Quadrant II to its final position in Quadrant IV. The overall composition of the transformations can be succinctly expressed as the rule: \[ (x, y) \rightarrow (x + 9, y - 9). \] This expression accurately reflects the net effect of both transformations. The transformations showcase the versatility of geometric manipulations and enhance our understanding of the coordinate plane's structure, allowing us to predict the outcomes of such transformations effectively. Through this analysis, we have gained a clearer picture of how objects can move and change within a mathematical framework, revealing the elegance and predictability inherent in geometric transformations.