Problem 95 - 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. $$g(x)=x^{3}-3$$

Step-by-Step Solution

Answer

To graph \(g(x) = x^{3} - 3\), first graph \(f(x)=x^{3}\), then shift the entire graph downward by 3 units.

Step 1 - Graph the Standard Cubic Function

Plot the graph of the cubic function \(f(x)=x^{3}\). The graph of \(f(x)=x^{3}\) starts from the bottom left at negative infinity, crosses the origin (0,0), and continues upwards to positive infinity, forming an 'S-like' shape.

Step 2 - Identify the Transformation

Notice that the function \(g(x) = x^{3} - 3\) is similar to the original function \(f(x)=x^{3}\), but with a '-3' at the end. This '-3' shifts the function downward by 3 units. This is called a vertical shift.

Step 3 - Perform the Transformation and Graph

Perform the vertical shift by subtracting '3' from the y-coordinates of the original function. Afterward, plot the graph for \(g(x) = x^{3} - 3\). It will look just like the original cubic function shifted down by 3 units.