Question

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 will follow the steps outlined in the question.

**Step 1: Apply the first transformation.**

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

We will apply this transformation to the vertices of ΔABC:

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

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

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

So the coordinates of ΔA'B'C' are:
- A'(-4, -2)
- B'(-1, 2)
- C'(1, -3)

**Step 2: Apply the second transformation.**

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

Now we apply this transformation to the new points:

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

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

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

So the coordinates of ΔA"B"C" are:
- A''(1, -5)
- B''(4, -1)
- C''(6, -6)

**Step 3: Combine the transformations into a single rule.**

To combine both transformations into one rule, we can add the changes together:

1. The x-coordinates change:
\((x + 4) + 5 = x + 9\)

2. The y-coordinates change:
\((y - 6) - 3 = y - 9\)

Thus, the composition transformation can be written as:
\[
(x, y) \to (x + 9, y - 9)
\]

**Step 4: Predict the quadrant.**

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

All the x-coordinates are positive, and all the y-coordinates are negative. This means that the points are located in the fourth quadrant, where x is positive and y is negative.

**Conclusion:**

The coordinates of ΔA"B"C" are:
- A''(1, -5)
- B''(4, -1)
- C''(6, -6)

The predicted quadrant for the image after the transformations is the **fourth quadrant**.