Question

find the coordinates after the following transformations. h(2, - 4) m(-3,5) p(5,4)

1. t<-2,5>(h)
2. r x - axis (p)
3. t(90°,0)(m)
4. t<4,-3>(p)
5. r y = 3(m)
6. t(180°,0)(h)

Explanation:

Step1: Translation of a point

For \(T\langle - 2,5
angle(H)\), if \(H=(2,-4)\), then the translation rule is \((x,y)\to(x - 2,y + 5)\).
Substitute \(x = 2\) and \(y=-4\) into the rule:
\(x'=2-2 = 0\), \(y'=-4 + 5=1\).

Step2: Reflection over the \(x\) - axis

For \(R_{x - axis}(P)\), if \(P=(5,4)\), the reflection rule over the \(x\) - axis is \((x,y)\to(x,-y)\).
Substitute \(x = 5\) and \(y = 4\) into the rule: \(x'=5\), \(y'=-4\).

Step3: Rotation of \(90^{\circ}\) counter - clockwise about the origin

For \(T_{(90^{\circ},0)}(M)\), if \(M=(-3,5)\), the rotation rule for \(90^{\circ}\) counter - clockwise about the origin is \((x,y)\to(-y,x)\).
Substitute \(x=-3\) and \(y = 5\) into the rule: \(x'=-5\), \(y'=-3\).

Step4: Translation of a point

For \(T\langle4,-3
angle(P)\), if \(P=(5,4)\), the translation rule is \((x,y)\to(x + 4,y-3)\).
Substitute \(x = 5\) and \(y = 4\) into the rule: \(x'=5 + 4=9\), \(y'=4-3 = 1\).

Step5: Reflection over the \(y\) - axis

For \(R_{y=3}(M)\), if \(M=(-3,5)\), the distance between \(y = 5\) and \(y = 3\) is \(d=5 - 3=2\). The reflection rule over the line \(y = 3\) is \((x,y)\to(x,3-(y - 3))=(x,6 - y)\).
Substitute \(x=-3\) and \(y = 5\) into the rule: \(x'=-3\), \(y'=6 - 5=1\).

Step6: Rotation of \(180^{\circ}\) about the origin

For \(T_{(180^{\circ},0)}(H)\), if \(H=(2,-4)\), the rotation rule for \(180^{\circ}\) about the origin is \((x,y)\to(-x,-y)\).
Substitute \(x = 2\) and \(y=-4\) into the rule: \(x'=-2\), \(y'=4\).

Answer:

1. \((0,1)\)
2. \((5,-4)\)
3. \((-5,-3)\)
4. \((9,1)\)
5. \((-3,1)\)
6. \((-2,4)\)