Responses
(x,y)→(−x,−y)
left parenthesis x comma y right parenthesis right arrow left parenthesis negative x comma negative y right parenthesis
(x,y)→(−y,x)
left parenthesis x comma y right parenthesis right arrow left parenthesis negative y comma x right parenthesis
(x,y)→(y,−x)
left parenthesis x comma y right parenthesis right arrow left parenthesis y comma negative x right parenthesis
(x,y)→(−y,−x)
9 answers
The mapping for a 270-degree (counterclockwise) rotation can be represented as:
\((x, y) \rightarrow (y, -x)\)
This means that the point \((x, y)\) after a 270-degree rotation becomes \((y, -x)\).
So, the correct answer is: \((x,y) \rightarrow (y,-x)\) (which corresponds to your third option).
To determine the degree and direction of rotation for the point \( P(-4, 10) \) rotated to \( P'(-10, -4) \), we can apply the mappings of rotation.
Using rotation mappings:
Now, let's analyze the transformation from \( P \) to \( P' \):
If we rotate by 90 degrees counterclockwise: \[ (x, y) \rightarrow (-y, x) \implies (-10, -4) \text{ (not equal to } P' \text{)}. \]
If we rotate by 180 degrees: \[ (x, y) \rightarrow (-x, -y) \implies (4, -10) \text{ (not equal to } P' \text{)}. \]
If we apply a 270-degree counterclockwise rotation (or equivalently a 90-degree clockwise rotation): \[ (x, y) \rightarrow (y, -x) \implies (10, 4). \] This is also not equal to \( P' \).
If we rotate by -90 degrees (clockwise): \[ (x, y) \rightarrow (-y, x) \implies (-10, -4) \text{ (this matches } P' \text{)}. \]
Thus, the correct answer is: −90 degrees (clockwise).
To find the coordinates of the image of a point \( Y \) after a 270-degree clockwise rotation about the origin, we can first understand that a 270-degree clockwise rotation is equivalent to a 90-degree counterclockwise rotation.
For a 90-degree counterclockwise rotation, the mapping for a point \( (x, y) \) transforms to: \[ (x, y) \rightarrow (-y, x) \]
Let’s apply this transformation to find the new coordinates of \( Y \):
The question seems to imply that \( Y \) has specific coordinates, but those aren't provided directly. Hence, let’s consider the potential answers:
To evaluate the options, we can check if any of these correspond to a valid translation from original points using the mapping rules. The transformation tells us that for any given point, if we have \( Y(x,y) \), its \( Y' \) after rotation would be created using the mapping.
So let's explore each choice:
To summarize, if you can provide the original coordinates of \( Y \), I can determine which of the options corresponds to a 270-degree clockwise rotation (or 90-degree counterclockwise). If no original coordinates are provided, none of the options can be confirmed as the image without that information.
To determine the new positions of the vertices \( P \) and \( R \) after the same rotation that moved \( Q(-4, -9) \) to \( Q'(-9, 4) \), we first need to identify the type of rotation that occurred.
Given the original point \( Q(-4, -9) \) and the rotated point \( Q'(-9, 4) \), we can find the rotation rule:
Let’s examine the transformations:
To understand what rotation this could correspond to, we enhance our view with the rotation mappings:
From the \( Q \) to \( Q' \) transformation, we can observe a 90-degree counterclockwise rotation:
Using the transformation for a 90-degree counterclockwise rotation, the mappings for \( P(-4, 0) \) and \( R(4, -4) \) are as follows:
For \( P(-4, 0) \): \[ P' = (0, -(-4)) = (0, -4) \]
For \( R(4, -4) \): \[ R' = (-(-4), 4) = (4, 4) \]
Now let's compare the images we calculated with the options provided:
Looking through your given responses, we find:
Thus, the correct responses should be P′(0,−4) and alternative R' to verify for vertices. The options hint at possible answers aligning with our mapping choice understanding.
Therefore, the correct interpretation corresponding closely is: P′(0,−4) and R′(−4,4).
The rotation function mapping for a 90-degree clockwise rotation about the origin is represented by the transformation:
\[ (x, y) \rightarrow (y, -x) \]
To find an equivalent transformation, let's consider the options provided:
A translation: This would simply move points without changing their orientation or shape, so this is not equivalent to a rotation.
A reflection over the line \( y = x \): The reflection over this line transforms the point \((x, y)\) to \((y, x)\), which is not equivalent to a 90-degree rotation.
A dilation with a scale factor of \( \frac{1}{2} \): Dilation changes the size of the figure but does not involve rotation, which makes it not equivalent.
A 270 degrees counterclockwise rotation: A 270-degree counterclockwise rotation can be expressed with the transformation: \[ (x, y) \rightarrow (-y, x) \]
This can also be related to a 90-degree clockwise rotation because:
Therefore, the correct equivalent transformation mapping for a 90-degree clockwise rotation is:
A 270 degrees counterclockwise rotation.
