Problem 106 - Precalculus Essentials

Question

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2}(x-2)^{3}-1$$

Step-by-Step Solution

Answer

The function \(h(x)=\frac{1}{2}(x-2)^3-1\) is a cubic function that is vertically stretched by a factor of 0.5, shifted 2 units to the right, and 1 unit downward.

Step 1: Graphing Standard Cubic Function

Start by graphing the standard cubic function \(f(x)=x^3\). This function gives a curve that passes through the origin (0,0), and it goes to positive infinity as \(x\) approaches positive infinity, and to negative infinity as \(x\) approaches negative infinity. It's crucial to understand this basic cubic curve as it will serve as the foundation for the transformation.

Step 2: Analyze Transformations

Next, identify the transformations present in the function \(h(x)=\frac{1}{2}(x-2)^3-1\). The function is altered by a vertical stretch by a factor of 0.5, a horizontal shift to the right by 2 units, and a vertical shift downward by 1 unit.

Step 3: Apply Transformations

Finally, apply the identified transformations to the graph of \(f(x)=x^3\). First, stretch the graph vertically by a factor of 0.5. This means that the curve will be closer to the x-axis than the original. Second, shift the graph 2 units to the right, which moves every point on the graph 2 units to the right. Lastly, shift the graph down by 1 unit, which moves every point on the graph 1 unit downward. The resulting graph will represent the function \(h(x)\).