Question

use transformations of the cube root function, ( f(x)=sqrt3{x} ), to graph the function ( g(x)=sqrt3{-x}-2 ).
choose the correct graph of ( g(x) ) below.
a.
b.
c.
d.

Explanation:

Step1: Analyze the transformation of the function

The parent function is \(y = \sqrt[3]{x}\). The function \(g(x)=\sqrt[3]{-x - 2}\) can be rewritten as \(g(x)=\sqrt[3]{-(x + 2)}\).
For a function \(y = f(x)\), the transformation \(y=f(-x)\) is a reflection about the \(y\) - axis, and the transformation \(y = f(x + h)\) (where \(h>0\)) is a horizontal shift to the left by \(h\) units.
For \(y=\sqrt[3]{x}\) to \(y=\sqrt[3]{-x}\), it is a reflection about the \(y\) - axis. Then for \(y=\sqrt[3]{-x}\) to \(y=\sqrt[3]{-(x + 2)}\), it is a horizontal shift to the left by \(2\) units.

Step2: Consider key points of the parent function

The parent function \(y=\sqrt[3]{x}\) has key points: when \(x = 0\), \(y = 0\); when \(x=1\), \(y = 1\); when \(x=-1\), \(y=-1\).
For the function \(y=\sqrt[3]{-x}\), when \(x = 0\), \(y = 0\); when \(x=-1\), \(y = 1\); when \(x = 1\), \(y=-1\).
For the function \(y=\sqrt[3]{-(x + 2)}\), let \(u=x + 2\), then \(y=\sqrt[3]{-u}\).
When \(x=-2\), \(y = 0\); when \(x=-3\), \(y = 1\); when \(x=-1\), \(y=-1\).

Answer:

B.