How do you describe the transformation of #f(x)=-absx-2# from a common function that occurs and sketch the graph?

Question

Serenity Johnson · Accepted Answer

You would describe two transformations of the graph: a reflection over the y-axis and a translation 2 units down.

This is what the graph looks like: graph{-abs(x)-2 [-10, 10, -10, 2]}

Chloe Alexander · Answer

undefined The transformation of the function $f(x) = -|x| - 2$ involves several key changes from the parent function $f(x) = |x|$. Firstly, there is a reflection about the x-axis due to the negative coefficient in front of the absolute value function. Secondly, there is a vertical translation downward by 2 units, as indicated by the constant term -2 added to the function.

To sketch the graph of $f(x) = -|x| - 2$, start with the parent function $f(x) = |x|$, which is a V-shaped graph centered at the origin. Then, apply the transformations: reflect the graph about the x-axis to obtain the downward opening V-shape, and shift the entire graph 2 units downward.

The resulting graph will resemble an upside-down V-shape with its vertex at (0, -2). The arms of the V will extend downward and the graph will continue indefinitely in both directions along the x-axis. Thus, the graph of $f(x) = -|x| - 2$ depicts a reflection of the absolute value function $|x|$ about the x-axis, followed by a vertical translation downward by 2 units.