Exercise 11 Page 173

Let's check each quadrant one by one.

Quadrant I

In Quadrant I, both $x$ and $y$ are positive numbers. Let's choose $(2,2)$ as our point. Then we multiply both coordinates by a negative number, let's use $\text{-}1.$ This reverses the point's signs so we get $(\text{-}2,\text{-}2).$ Let's plot these points.

Quadrant II

In Quadrant II, $x$ is negative and $y$ is positive. Let's choose $(\text{-}1,1)$ as our point. Then we multiply both coordinates by a negative number, let's use $\text{-}1.$ This reverses the point's signs so we get $(1,\text{-}1).$ Let's plot these points.

Quadrant III

In Quadrant III, both $x$ and $y$ are negative numbers. Let's choose $(\text{-}3,\text{-}3)$ as our point. Then we multiply both coordinates by a negative number, let's use $\text{-}1.$ This reverses the point's signs so we get $(3,3).$ Let's plot these points.

Quadrant IV

In Quadrant IV, $x$ is positive and $y$ is negative. Let's choose $(4,\text{-}4)$ as our point. Then we multiply both coordinates by a negative number, let's use $\text{-}1.$ This gives the point $(\text{-}4,4).$ Let's plot these points.

Conclusion

In each case, multiplying the point by a negative number carries the point diagonally across the origin to the new point. This is called a reflection about the origin.