Let's start out with the parent function $y = ∣ x ∣$ which has its vertex at $(0, 0) .$

Examining the calculator window, we see that its plot is upside down compared to the parent function. To turn the parent function upside down, we have to reflect it in the

x -

axis. This can be done by multiplying the absolute value by

- 1 :

y = - 1 \cdot ∣ x ∣ \Leftrightarrow y = - ∣ x ∣ .

Let's show the reflection of the parent function.

The plot from the calculator reaches its vertex at

(- 1, 1) .

To move the vertex from

(0, 0)

to this location, we have to translate the graph vertically one step up by adding

1

to the function:

y = - ∣ x ∣ + 1 .

Finally, we have to translate the function horizontally one step to the left. To do that we add

1

to

x :

y = - ∣ x + 1 ∣ + 1 .

Let's show these two translations.

This does resemble the plot in our graphing calculator. Let's plot it for ourselves to make sure. To enter the equation, we press $Y =$ on our calculator, and then write the equation on one of the rows.

Next, by pressing GRAPH, the calculator will plot the equation.

The calculator displays the standard viewing window. Let's change it to $- 6 \leq x \leq 6$ and $- 4 \leq y \leq 4$ and press GRAPH again to plot the equation.

We are correct!

Exercise