Problem 96 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of the given function \(g(x) = x^{3} - 2\) will have the same 'sweeping' shape as the standard cubic function \(f(x) = x^{3}\), but it will be shifted downward by 2 units.

Step 1 - Graph the standard cubic function

Start with the standard cubic function \(f(x)=x^{3}\). Plot some points like (-2, -8), (-1, -1), (0, 0), (1, 1) and (2, 8) to draw the curve. The graph should cross through the origin (0,0) and continue to increase as x becomes more positive and decreases as x becomes more negative. Note the general shape of the graph, how it 'sweeps' across the four quadrants.

Step 2 - Analyze the transformations

The given function to graph is \(g(x) = x^{3} - 2\). This function indicates that there's a vertical shift downward by 2 units from our original function \(f(x) = x^{3}\). This transformation will shift every point on the original graph downward by 2 units.

Step 3 - Graph the transformed function

Using the shape noted from the graph of \(f(x)\), plot the transformation of \(g(x)\). Shift each point from the original function downward by two units due to the -2 in the function \(g(x) = x^{3} - 2\). Just as before, the shape of the graph should remain the same, yet will now be shifted down 2 units.

Key Concepts

Standard Cubic Function

The standard cubic function is represented by the equation \( f(x) = x^3 \). It's the simplest form of cubic functions and serves as the foundation for understanding more complex cubic equations. The graph of a standard cubic function has distinct features: it passes through the origin \( (0,0) \), and as the value of \( x \) increases or decreases, the function increases or decreases rapidly due to the cubic nature of the equation.

To plot the standard cubic function, you would calculate the cube of a selection of positive and negative values of \( x \), and then plot these points on a graph to see the smooth curve that is symmetric with respect to the origin. It's important to note that it will pass through all four quadrants, showing a distinct 'S' shape as it goes.

Transformations of Functions

Transformations of functions involve changing the position, shape, and size of graphs without changing their overall characteristics. The most common transformations include vertical and horizontal shifts, stretches and compressions, and reflections over the axes.

Understanding these transformations is vital to graph more complex functions efficiently. In the context of a cubic function, transformations allow us to take the standard graph \( f(x) = x^3 \) and alter it to fit different equations like \( g(x) = x^3 - 2 \), which involves a vertical shift. By applying these transformations step by step, one can graph complex functions using the simpler standard graph as a starting point.

Vertical Shift

A vertical shift occurs when every point of a function's graph is moved up or down the y-axis by the same amount. In the equation \( g(x) = x^3 - 2 \), the term '-2' indicates a vertical shift of two units downwards.

When graphing the function, after plotting the points of the standard cubic function, you would simply shift each of those points down by 2 units to represent the new function. The vertical shift doesn't change the shape of the graph, it merely translates the entire graph up or down depending on the sign and magnitude of the vertical shift.

Plotting Points

Plotting points is an essential starting step in graphing functions. By selecting a series of x-values, computing the corresponding y-values, and then marking those \((x, y)\) points on the graph, you get a set of landmarks from which you can trace the curve of the function.

For cubic functions like the standard cubic function, choosing a range of x-values that includes both negative and positive values will help illustrate the way the function behaves across different quadrants of the graph. Once you have marked these points, you can draw a smooth curve through them to complete the graph of the function.