Exercise 48 Page 161

We want to write an absolute value function whose graph forms a square with the given graph. To do so, we will need to draw any horizontal line. Let's arbitrarily choose the line $y=3.$

Note that, due to the symmetry of the absolute value function, we have an isosceles triangle. Since one of the angles is a right angle, the measure of the other two angles must be $45^{\circ }.$ Let's now translate our graph down $3$ units. We obtain this transformation by subtracting $3$ units from the output. Now the triangle will be formed with the $x\text{-}$axis. Let's now reflect our graph in the $x\text{-}$axis. We obtain this by changing the sign of the output. Let's graph the above function.

Note that the triangles above are congruent. Therefore, the obtained triangle is an isosceles triangle with one right angle and two angles whose measure is $45^{\circ }.$ If we merge these isosceles triangles, we obtain a square.

Finally, we have to go back to the original position. To do so, we will translate both graphs up $3$ units by adding $3$ to the outputs. Let's see the final graph.

Note that there are infinitely many functions whose graphs form a square with the given graph. The function we found, $y=\text{-}|x-3|+5,$ is just one of them.