A figure is rotated 90 degrees counterclockwise about the origin. Which of the following function mappings was applied? Enter the number of the correct option.
To determine the correct function mapping for rotating a figure 90 degrees counterclockwise about the origin, we need to analyze the effect of the rotation on the coordinates of the original figure.
When a figure is rotated counterclockwise about the origin, the x-coordinate becomes the new y-coordinate, and the y-coordinate becomes the negated x-coordinate.
Therefore, the correct function mapping for rotating a figure 90 degrees counterclockwise about the origin is:
f(x, y) = (-y, x)
So, the correct option is 2.
To determine the correct option, we need to understand how the point (x, y) is rotated counterclockwise 90 degrees about the origin.
When a point is rotated counterclockwise 90 degrees about the origin, the x-coordinate becomes the new y-coordinate, and the y-coordinate becomes the new negative x-coordinate.
So, for a point (x, y) rotated 90 degrees counterclockwise, the new point would be (-y, x).
Now let's look at the options and try applying this rotation to each one:
1. f(x) = x^2 - y^2
If we apply the rotation to (x, y), we would get (-y, x) as explained before. But this option does not match that pattern.
2. f(x) = x^2 + y^2
If we apply the rotation to (x, y), we would get (-y, x) as explained before. This option matches the pattern, so it could be the correct mapping.
3. f(x) = x^2 - 2xy + y^2
If we apply the rotation to (x, y), we would get (-y, x) as explained before. But this option does not match that pattern.
4. f(x) = -x^2 + 2xy + y^2
If we apply the rotation to (x, y), we would get (-y, x) as explained before. But this option does not match that pattern.
From the analysis, it appears that option 2, f(x) = x^2 + y^2, matches the function mapping for rotating a figure 90 degrees counterclockwise about the origin. Therefore, the correct option is 2.