Question

1. write a sentence describing the transformation rule: $(x,y)\to(x - 7,y + 3)$
2. what type of transformation moves a figure left, right, up or down?
3. pont $x(-3,-2)$ is translated using the rule $(x,y)\to(x + 3,y + 5)$, then reflect x - axis. what are the coordinates of $x$?
4. describe in your own words what happens to the x & y coordinates whe

(a) $90\degree$ counterclockwise?
b). rotate them 180 degrees?

Explanation:

Step1: Apply translation rule

For the translation rule \((x,y)\to(x - 7,y + 3)\), the \(x\)-coordinate of each point is decreased by \(7\) (which means a horizontal shift) and the \(y\)-coordinate is increased by \(3\) (which means a vertical shift).
The transformation is a translation \(7\) units to the left (because \(x=x-7\)) and \(3\) units up (because \(y=y + 3\)).

Step2: Identify transformation type (Question 15)

A translation (or slide) is the type of transformation that moves a figure left, right, up or down.

Step3: Find coordinates after translation and reflection (Question 16)

• Translation:

Given the point \(X(-3,-2)\) and the translation rule \((x,y)\to(x + 3,y + 5)\).
For the \(x\)-coordinate: \(x=-3+3 = 0\)
For the \(y\)-coordinate: \(y=-2 + 5=3\)
After translation, the point \(X'\) has coordinates \((0,3)\)

• Reflection over the \(x\)-axis:

The rule for reflection over the \(x\)-axis is \((x,y)\to(x,-y)\)
For the point \(X'(0,3)\), after reflection over the \(x\)-axis, the \(x\)-coordinate remains \(0\) and the \(y\)-coordinate becomes \(-3\). So \(X''=(0,-3)\)

Step4: Describe rotation effects (Question 17)

• a) \(90^{\circ}\) counter - clockwise rotation:

If we have a point \((x,y)\) and rotate it \(90^{\circ}\) counter - clockwise about the origin, the new coordinates \((x',y')\) follow the rule \((x,y)\to(-y,x)\). The \(x\) and \(y\) coordinates are swapped and the new \(x\) - coordinate is the negative of the original \(y\) - coordinate.

• b) \(180^{\circ}\) rotation:

If we rotate a point \((x,y)\) \(180^{\circ}\) about the origin, the new coordinates \((x',y')\) follow the rule \((x,y)\to(-x,-y)\). Both the \(x\) and \(y\) coordinates change their signs.

Answer:

1. The transformation is a translation \(7\) units to the left and \(3\) units up.
2. Translation
3. \((0,-3)\)

17.
a) The \(x\) and \(y\) coordinates are swapped and the new \(x\) - coordinate is the negative of the original \(y\) - coordinate.
b) Both the \(x\) and \(y\) coordinates change their signs.