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Use the image to answer the question.

Five solid lines join five plotted points labeled upper I, upper F, upper Y, upper H, and upper S on a coordinate plane. An unlabeled point is also plotted on the plane.
cordinates: F: -6, 1. Y, -9, -1. H: -12, 1. S: -12, 4, and I, -8, 4. the unlabeled point is on 2 on the vertical axis, somehwere beween -8 and -7 on the horizontal axis.
Image Long DescriptionThe values on the x-axis range from negative 15 to 5 in one-unit increments. The values on the y-axis range from negative 5 to 10 in one-unit increments. The coordinates of the plotted points are as follows: upper I at left parenthesis negative 8 comma 4 right parenthesis, upper F at left parenthesis negative 6 comma 1 right parenthesis, upper Y at left parenthesis negative 9 comma negative 1 right parenthesis, upper H at left parenthesis negative 12 comma 1 right parenthesis, and upper S at left parenthesis negative 12 comma 4 right parenthesis. Solid lines join upper I to upper F, upper F to upper Y, upper Y to upper S, upper S to upper H, and upper H to upper I. An unlabeled point is plotted at approximately left parenthesis negative 7.5 comma 2 right parenthesis.

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.

(1 point)
Responses

Y′(1, 9)
upper Y prime left parenthesis 1 comma space 9 right parenthesis

Y′(1, −9)
upper Y prime left parenthesis 1 comma space negative 9 right parenthesis

Y′(−9, 1)
upper Y prime left parenthesis negative 9 comma space 1 right parenthesis

Y′(−1, 9)

1 answer

To determine the new coordinates of point Y after a -270-degree rotation about the origin, we can use the rotation transformation formulas for a point \((x, y)\).

The rotation formulas for a \(-\theta\) degree rotation about the origin are:

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

For a \(-270\) degree rotation, it is equivalent to a \(90\) degree rotation (since \(-270\) + \(360\) = \(90\)). The cosine and sine values for \(90\) degrees are:

  • \(\cos(90^\circ) = 0\)
  • \(\sin(90^\circ) = 1\)

Thus, the transformation equations for a \(90\) degree rotation around the origin modify to:

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

Now we apply this to point Y, which has coordinates \((-9, -1)\):

  1. Calculate \(x'\): \[ x' = -(-1) = 1 \]

  2. Calculate \(y'\): \[ y' = -9 \]

Thus, the new coordinates \(Y' = (1, -9)\).

Therefore, the coordinates of the image of \(Y\) after a \(-270^\circ\) rotation about the origin are:

\[ Y' \left( 1, -9 \right) \]