Question

A fish was plotted on a coordinate plane. Identify the coordinates of the image of Y that would create a -270- degree rotation about the origin. A: Y'(1,-9) B:y'(-1,9) C:Y'(-9,1) D:y'(1,9)

Answer 1

To rotate a point \( (x, y) \) by an angle \( \theta \) about the origin in a coordinate plane, you can use the following formulas:

\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]

For a rotation of \( -270^\circ \), we convert it to radians:

\[
-270^\circ = -\frac{3\pi}{2} \text{ radians}
\]

Using the cosine and sine values:

\[
\cos(-270^\circ) = 0, \quad \sin(-270^\circ) = -1
\]

Applying the rotation formulas gives us:

\[
x' = x \cdot 0 - y \cdot (-1) = y
\]
\[
y' = x \cdot (-1) + y \cdot 0 = -x
\]

So a point \( (x, y) \) after a \( -270^\circ \) rotation will transform to:

\[
(x', y') = (y, -x)
\]

Now, let's apply this to find the coordinates of \( Y' \):

Assuming \( Y \) has coordinates \( (x, y) \):
- The new \( x' \) (the first coordinate) will be the old \( y \).
- The new \( y' \) (the second coordinate) will be the negation of the old \( x \).

Now let's check the options:

A: \( Y'(1, -9) \) would correspond to some \( (x, y) \) making \( y = 1 \) and \( -x = -9 \), hence \( x = 9 \). This corresponds with \( (9, 1) \).

B: \( Y'(-1, 9) \) gives \( y = -1 \) and \( -x = 9 \), thus \( x = -9 \). This corresponds with \( (-9, -1) \).

C: \( Y'(-9, 1) \) gives \( y = -9 \) and \( -x = 1 \), thereby \( x = -1 \). This corresponds with \( (-1, -9) \).

D: \( Y'(1, 9) \) gives \( y = 1 \) and \( -x = 9 \); thus \( x = -9 \). This corresponds with \( (-9, 1) \).

The correct rotated coordinate option (assuming the original coordinates for \( Y \) are \( (9, 1) \)) would be A: \( Y'(1, -9) \).

Note that in general you need the original point \( Y \) to get an exact transformation path. Here, we determined possibilities based on rotation mathematics. Depending on which point was \( Y \), any of the transformations across your options could be valid based on different starting \( (x, y) \) pairs. Most likely for the intended rotation \( -270^\circ \):

C: \( Y'(-9, 1) \) is actually what we want to associate it with \( (1, 9) \).