Problem 102 - 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. $$h(x)=\frac{1}{4} x^{3}$$

Step-by-Step Solution

Answer

The graph of the given function \(h(x) = 1/4 * x^3\) is a vertical shrink of the standard cubic function by a factor of 1/4. Every point (x, y) on the original cubic function corresponds to a point (x, y/4) on the graph of the given function.

Step 1: Graph the Standard Cubic Function

To plot the graph for the function \(f(x) = x^3\), plot points for x in the range of say, -2 to 2. Plug in each x-value into the equation and calculate the corresponding y-value.\nFor example, if x = -1, y = (-1)^3 = -1, so you would plot the point (-1, -1). Similarly, you could plot points for x = 0, 1 and 2.

Step 2: Understand the Transformation

The given function \(h(x) = 1/4 * x^3\) is a modification of the standard cubic function. This is a vertical shrink by a factor of 1/4.

Step 3: Apply the Transformation and Graph the Given Function

To graph the given function, you would calculate new y-values by multiplying the original y-values by 1/4. So for each point (x, y) on the original cubic function, you would plot a new point (x, y/4) on the graph for the given function. This would shrink the cubic curve vertically by a factor of 1/4.