Question

1. write an algebraic formula for a rotation in which the preimage coordinates are a(1, -3), b(1, 1), c(5, 1) and the image points are a(-1, 3), b(-1, -1), c(-5, -1).
2. w(5, 6) is the result of a 90° counter - clockwise rotation. find the coordinates of point w.
3. the points d(0, 0), e(1, 3), f(-4, 5) represent the vertices of a polygon. find the coordinates of the image polygon after a rotation modeled by the rule (x, y) → (-y, x).
4. given the preimage coordinate d(7, 2) and an image point d(2, -7) describe the counter - clockwise rotation and write an algebraic formula for the rotation that takes place on point d.
5. triangle abc is rotated 360° resulting in image points a(-5, 5), b(3, 1) and c(4, -4). what is the coordinate of preimage point b?
6. the line segment (overline{rs}) with endpoints r(9, 3) and s(3, 6) is rotated so the r is located at (-9, -3). write an algebraic rule that represents the counter - clockwise rotation and find s.
7. find the image point for m((\frac{1}{2}), -5) after a 270° counterclockwise rotation.
8. the points d(0, 0), e(1, 3), f(-4, 5) represent the image vertices of a polygon. find the coordinates of the preimage polygon after a 270° counter - clockwise rotation.
9. given the algebraic rule (x, y) = (-x, -y) and the point e(7, -2), find e.

Explanation:

13.

Step1: Analyze coordinate - change pattern

Observe the pre - image points \(A(1, - 3)\) and \(A'(-1,3)\), \(B(1,1)\) and \(B'(-1,-1)\), \(C(5,1)\) and \(C'(-5,-1)\). We can see that \((x,y)\to(-x,-y)\).

Step1: Recall 90 - degree counter - clockwise rotation rule

The rule for a 90 - degree counter - clockwise rotation is \((x,y)\to(-y,x)\). Let the coordinates of \(W\) be \((x,y)\), then \((-y,x)=(5,6)\). So, \(-y = 5\) and \(x = 6\), which gives \(y=-5\) and \(x = 6\).

Step1: Apply the rotation rule \((x,y)\to(-y,x)\) to each point

For point \(D(0,0)\): \((0,0)\to(0,0)\)
For point \(E(1,3)\): \((1,3)\to(-3,1)\)
For point \(F(-4,5)\): \((-4,5)\to(-5,-4)\)

Answer:

The algebraic formula for the rotation is \((x,y)\to(-x,-y)\)

14.