Problem 115 - Precalculus Essentials

Question

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt[3]{x+2}$$

Step-by-Step Solution

Answer

The cube root graph \(f(x)\) is transformed into the graph of \(h(x)\) through a shift of 2 units to the left and a reflection in the x-axis.

Step 1: Graph the Cube Root Function

To graph \(f(x)=\sqrt[3]{x}\), plot a point at the origin (0,0), because the cube root of 0 is 0. Then plot points for each integer value of x, with y being the cube root of x. Connect the points to create a continuous curve.

Step 2: Understand the Transformations

Looking at \(h(x) = -\sqrt[3]{x+2}\), there are two transformations from the original function \(f(x)\). The +2 inside the cube root shifts the graph 2 units to the left, and the negative sign in front of the cube root reflects the graph in the x-axis.

Step 3: Apply the Transformations on the Graph of the Cube Root Function

To graph \(h(x)\), take the original graph of \(f(x)\), shift it 2 units to the left, and then reflect it over the x axis. The outcome will be the graph of \(h(x)= -\sqrt[3]{x+2}\).