Question

(-8, 1) over the y-axis (8,1); (-8,6) over the line y = -x; (8, -6) over the line y = x; (-3, -9) over the x-axis; (3,7) ov; (-2, -8) o; (-4, -5) ov; (0, -6) over

Explanation:

Problem 6: Reflect \((-8, 6)\) over the line \(y = -x\)

Step 1: Recall the reflection rule over \(y = -x\)

The rule for reflecting a point \((x, y)\) over the line \(y=-x\) is to swap the \(x\) and \(y\) coordinates and then change their signs. Mathematically, if we have a point \((x,y)\), its reflection over \(y = -x\) is \((-y,-x)\).

Step 2: Apply the rule to the point \((-8, 6)\)

For the point \((x=-8,y = 6)\), we first swap the coordinates: \((6,-8)\), and then change the signs of both coordinates. So the \(x\)-coordinate of the new point is \(-6\) and the \(y\)-coordinate is \(8\). So the reflected point is \((-6,8)\)? Wait, no, wait. Wait the rule is: If \((x,y)\) is reflected over \(y=-x\), the image is \((-y,-x)\). Let's verify:

Let \(x=-8\) and \(y = 6\). Then \(-y=-6\) and \(-x = 8\). So the reflected point is \((-y,-x)=(-6,8)\)? Wait, no, let's take a simple example. Let's take the point \((1,1)\) reflected over \(y=-x\). The line \(y = -x\) has a slope of \(- 1\) and passes through the origin. The reflection of \((1,1)\) over \(y=-x\) should be \((-1,-1)\). Using the formula \((-y,-x)\) where \(x = 1,y=1\), we get \((-1,-1)\), which is correct. Another example: \((2,3)\) reflected over \(y=-x\) should be \((-3,-2)\). Using the formula \((-y,-x)\) with \(x = 2,y = 3\), we get \((-3,-2)\), which is correct. So for the point \((-8,6)\), \(x=-8\), \(y = 6\). Then \(-y=-6\) and \(-x=8\). So the reflected point is \((-y,-x)=(-6,8)\)? Wait, no, wait \((x,y)=(-8,6)\), so swapping \(x\) and \(y\) gives \((6,-8)\), then multiplying both by \(- 1\) gives \((-6,8)\). Yes, that's correct.

Wait, let's do it geometrically. The line \(y=-x\) is a line that makes an angle of \(135^{\circ}\) with the positive \(x\)-axis. The reflection of a point \((x,y)\) over \(y=-x\) can be found by the transformation: \((x,y)\to(-y,-x)\). So for \((-8,6)\), applying the transformation: \(x=-8\), \(y = 6\), so the new \(x\) is \(-y=-6\), new \(y\) is \(-x = 8\). So the reflected point is \((-6,8)\).

Problem 7: Reflect \((8, -6)\) over the line \(y = x\)

Step 1: Recall the reflection rule over \(y = x\)

The rule for reflecting a point \((x, y)\) over the line \(y = x\) is to swap the \(x\) and \(y\) coordinates. Mathematically, if we have a point \((x,y)\), its reflection over \(y=x\) is \((y,x)\).

Step 2: Apply the rule to the point \((8, -6)\)

For the point \((x = 8,y=-6)\), swapping the \(x\) and \(y\) coordinates gives \((-6,8)\). Let's verify with an example. The point \((2,3)\) reflected over \(y=x\) is \((3,2)\), which is correct. So for \((8,-6)\), swapping \(x\) and \(y\) gives \((-6,8)\).

Problem 8: Reflect \((-3, -9)\) over the \(x\)-axis

Step 1: Recall the reflection rule over the \(x\)-axis

The rule for reflecting a point \((x, y)\) over the \(x\)-axis is to keep the \(x\)-coordinate the same and change the sign of the \(y\)-coordinate. Mathematically, if we have a point \((x,y)\), its reflection over the \(x\)-axis is \((x,-y)\).

Step 2: Apply the rule to the point \((-3, -9)\)

For the point \((x=-3,y = -9)\), the \(x\)-coordinate remains \(-3\) and the \(y\)-coordinate becomes \(-(-9)=9\). So the reflected point is \((-3,9)\). Let's verify with an example. The point \((2,3)\) reflected over the \(x\)-axis is \((2,-3)\), which is correct. So for \((-3,-9)\), applying the rule gives \((-3,9)\).

Answer:

s:

• Problem 6: \(\boldsymbol{(-6, 8)}\)
• Problem 7: \(\boldsymbol{(-6, 8)}\) (Wait, no, wait for problem 7, \((8,-6)\) over \(y = x\) is \((-6,8)\)? Wait, no, \((x,y)=(8,-6)\), swapping \(x\) and \(y\) gives \((-6,8)\), yes.
• Problem 8: \(\boldsymbol{(-3, 9)}\)