Problem 116 - 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

To graph the function \(h(x)=-\sqrt[3]{x-2}\), we first graph the cube root function \(f(x)=\sqrt[3]{x}\), then apply a horizontal shift 2 units to the right and finally reflect the graph across the x-axis.

Step 1: Graph the Cube Root Function \(f(x)=\sqrt[3]{x}\)

Begin by sketching the graph of the cube root function, \(f(x)=\sqrt[3]{x}\). This function starts at negative infinity, crosses the origin (0,0) and then heads to positive infinity. It basically looks like the right half of a sideways parabola.

Step 2: Apply Horizontal Shift

To graph the function \(h(x)=-\sqrt[3]{x-2}\), first apply the horizontal shift. Every x-value in the original function \(f(x)\) is increased by 2. This means that our graph will shift 2 units to the right.

Step 3: Apply Vertical Reflection

The negative sign in front of the cube root in the function \(h(x)=-\sqrt[3]{x-2}\) corresponds to a reflection of the graph across the x-axis. This means every positive y-value from the cube root function will become negative and vice versa.