Question

1. d(2,9) is the image of d after a reflection in the y - axis. what are the coordinates of d? d(__,) 7. q(5, - 1) is the image of q after a reflection in the line y = - x. what are the coordinates of q? q(,) 8. give the coordinates of l(-1,5) after a 270° clockwise rotation about the origin. l(,) 9. n(-10, - 15) is the image of n after a dilation with a scale factor of 5/4, centered at the origin. what are the coordinates of n? n(,__) 10. what is the rule for a 90° rotation (counterclockwise)? (x,y)→(-x, - y) (x,y)→(y, - x) (x,y)→(-y,x)

Explanation:

6. Reflection in the y - axis

Step1: Rule for reflection in the y - axis

The rule for reflecting a point \((x,y)\) in the \(y\) - axis is \((x,y)\to(-x,y)\). Let \(D=(x,y)\) and \(D'=(2,9)\).

Step2: Apply the rule

If \(-x = 2\) and \(y = 9\), then \(x=-2\).

7. Reflection in the line \(y=-x\)

Step1: Rule for reflection in the line \(y =-x\)

The rule for reflecting a point \((x,y)\) in the line \(y=-x\) is \((x,y)\to(-y,-x)\). Let \(Q=(x,y)\) and \(Q'=(5,-1)\).

Step2: Apply the rule

If \(-y = 5\) and \(-x=-1\), then \(y=-5\) and \(x = 1\).

8. \(270^{\circ}\) clockwise rotation about the origin

Step1: Rule for \(270^{\circ}\) clockwise rotation

The rule for a \(270^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y,-x)\). Given \(L=(-1,5)\).

Step2: Apply the rule

Substitute \(x=-1\) and \(y = 5\) into the rule: \(x'=5\) and \(y'=1\).

9. Dilation with scale factor \(k=\frac{5}{4}\) centered at the origin

Step1: Rule for dilation

The rule for dilation centered at the origin with scale factor \(k\) is \((x,y)\to(kx,ky)\). Let \(N=(x,y)\) and \(N'=(-10,-15)\) and \(k=\frac{5}{4}\). Then \(kx=-10\) and \(ky=-15\).

Step2: Solve for \(x\) and \(y\)

If \(\frac{5}{4}x=-10\), then \(x=-10\times\frac{4}{5}=-8\); if \(\frac{5}{4}y=-15\), then \(y=-15\times\frac{4}{5}=-12\).

10. \(90^{\circ}\) counter - clockwise rotation

Step1: Recall rotation rules

For a \(90^{\circ}\) counter - clockwise rotation about the origin, if we start with a point \((x,y)\), the new point \((x',y')\) is given by \((x,y)\to(-y,x)\).

Answer:

1. \(D(-2,9)\)
2. \(Q(1,-5)\)
3. \(L'(5,1)\)
4. \(N(-8,-12)\)
5. \((x,y)\to(-y,x)\) (the third option)