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

4 5^{\circ} .

Let's now translate our graph down

3

units. We obtain this transformation by subtracting

3

units from the output.

y = ∣ x - 3 ∣ + 1 - 3 \Leftrightarrow y = ∣ x - 3 ∣ - 2

Now the triangle will be formed with the

x -

axis.

Let's now reflect our graph in the

x -

axis. We obtain this by changing the sign of the output.

y = - (∣ x - 3 ∣ - 2) \Leftrightarrow y = - ∣ x - 3 ∣ + 2

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 $4 5^{\circ} .$ 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.

y = ∣ x - 3 ∣ - 2 + 3 ⇕ y = ∣ x - 3 ∣ + 1 y = - ∣ x - 3 ∣ + 2 + 3 ⇕ y = - ∣ x - 3 ∣ + 5

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, $y = - ∣ x - 3 ∣ + 5,$ is just one of them.

Exercise