Question

which two transformations could have been used to create figure 2?

drag and drop the correct transformations into the boxes.

first, figure 1 was

then, it was to create figure 2.

options:

• translated down 7 units
• reflected across the \\(x\\)-axis
• rotated \\(90^\circ\\) clockwise around the origin
• rotated \\(90^\circ\\) counterclockwise around the origin

Explanation:

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<reasoning>

Identify key vertices of Figure 1 and Figure 2

\[

$$\begin{aligned} &\text{Figure 1 vertices: } A(-5, 2), B(-3, 2), C(-2, 5), D(-4, 5), E(-4.5, 3.5) \\ &\text{Figure 2 vertices: } A'(2, -5), B'(2, -3), C'(5, -2), D'(5, -4), E'(3.5, -4.5) \end{aligned}$$

\]

Test transformation sequences

\[

$$\begin{aligned} &\text{Let } (x, y) \text{ be a point on Figure 1.} \\ &\text{If reflected across the } x\text{-axis: } (x, y) \to (x, -y) \\ &\text{Applying to } A(-5, 2) \to A_1(-5, -2) \\ &\text{If then rotated } 90^\circ \text{ counterclockwise around the origin: } (x_1, y_1) \to (-y_1, x_1) \\ &\text{Applying to } A_1(-5, -2) \to A'(-(-2), -5) = (2, -5) \\ &\text{Applying to } C(-2, 5) \to C_1(-2, -5) \to C'(-(-5), -2) = (5, -2) \end{aligned}$$

\]

Verify matching with options

\[

$$\begin{aligned} &\text{First transformation: "reflected across the } x\text{-axis"} \\ &\text{Second transformation: "rotated } 90^\circ \text{ counterclockwise around the origin"} \end{aligned}$$

\]
</reasoning>

<answer>
First, figure 1 was <blank>reflected across the \(x\)-axis</blank>

Then, it was <blank>rotated \(90^\circ\) counterclockwise around the origin</blank> to create figure 2.
</answer>

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Answer:

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<reasoning>

Identify key vertices of Figure 1 and Figure 2

\[

$$\begin{aligned} &\text{Figure 1 vertices: } A(-5, 2), B(-3, 2), C(-2, 5), D(-4, 5), E(-4.5, 3.5) \\ &\text{Figure 2 vertices: } A'(2, -5), B'(2, -3), C'(5, -2), D'(5, -4), E'(3.5, -4.5) \end{aligned}$$

\]

Test transformation sequences

\[

$$\begin{aligned} &\text{Let } (x, y) \text{ be a point on Figure 1.} \\ &\text{If reflected across the } x\text{-axis: } (x, y) \to (x, -y) \\ &\text{Applying to } A(-5, 2) \to A_1(-5, -2) \\ &\text{If then rotated } 90^\circ \text{ counterclockwise around the origin: } (x_1, y_1) \to (-y_1, x_1) \\ &\text{Applying to } A_1(-5, -2) \to A'(-(-2), -5) = (2, -5) \\ &\text{Applying to } C(-2, 5) \to C_1(-2, -5) \to C'(-(-5), -2) = (5, -2) \end{aligned}$$

\]

Verify matching with options

\[

$$\begin{aligned} &\text{First transformation: "reflected across the } x\text{-axis"} \\ &\text{Second transformation: "rotated } 90^\circ \text{ counterclockwise around the origin"} \end{aligned}$$

\]
</reasoning>

<answer>
First, figure 1 was <blank>reflected across the \(x\)-axis</blank>

Then, it was <blank>rotated \(90^\circ\) counterclockwise around the origin</blank> to create figure 2.
</answer>

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