Problem 9 - Technical Mathematics with Calculus

Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=x^{3}-2$$

Step-by-Step Solution

Answer

Plot points such as \((-2, -10)\), \((-1, -3)\), \((0, -2)\), \((1, -1)\), and \((2, 6)\). Then, draw a smooth curve through these points to graph \(y = x^3 - 2\). The initial window of \(-5\) to \(5\) for \(x\) and \(y\) should suffice.

Step 1: Identify the Function

First, recognize the function that needs to be graphed: the cubic function given by the equation \(y = x^3 - 2\). This is a transformation of the basic cubic function \(y = x^3\), shifted downward by 2 units.

Step 2: Plot Basic Points

Start plotting points that satisfy the equation on a graph. Since the function is symmetric for positive and negative values of \(x\), it is sufficient to find points for \(x\) from \(-5\) to \(5\). List some key points such as \((-2, -10)\), \((-1, -3)\), \((0, -2)\), \((1, -1)\), \((2, 6)\). This provides a rough shape of the graph.

Step 3: Draw the Graph

Using the plotted points, start drawing a smooth curve that passes through these points. Remember the end behavior of the cubic function: as \(x\) approaches negative infinity, \(y\) also approaches negative infinity, and as \(x\) approaches positive infinity, \(y\) approaches positive infinity.

Step 4: Adjust the Viewing Window

With the initial window set from \(-5\) to \(5\), check if the curve is correctly displayed within this window. If any part of the graph is missing, adjust the window accordingly. For \(y = x^3 - 2\), the initial window will likely be sufficient.

Key Concepts

Cubic Function Transformation

Understanding the transformation of cubic functions is crucial when graphing. The standard cubic function has the form \(y = x^3\). However, by adding or subtracting values, we shift the graph vertically or horizontally. In our example, \(y = x^3 - 2\), the \( -2\) indicates a vertical downward shift of the graph by two units. This small tweak can significantly alter the appearance of the graph. It is imperative to recognize this shift to accurately plot the graph's new position on the coordinate system.

To visualize these transformations, imagine the basic graph of \(y = x^3\) sliding down along the y-axis. Every point on the graph moves down uniformly, and this movement does not affect the shape or orientation of the original cubic function, which retains its distinctive 'S' shape even after the transformation.

Plotting Points in Coordinate System

Plotting points is a foundational step in graphing functions. When graphing \(y = x^3 - 2\), start by selecting a range of x-values. For this function, using values from \(x = -5\) to \(x = 5\) is a good starting point. Calculate the corresponding y-values using the function formula for each chosen x-value.

For instance, when \(x = -2\), we calculate \(y = (-2)^3 - 2 = -10\). Repeat this process for other values of x to get a set of points such as \( (-2, -10)\), \( (-1, -3)\), \( (0, -2)\), \( (1, -1)\), and \( (2, 6)\). Plotting these points provides a scaffold for the graph. Connect these points with a smooth curve, considering the nature of cubic functions to predict the overall shape properly.

Symmetry of Functions

Symmetry in functions can simplify graphing, especially with functions like cubic functions that display point symmetry about their inflection point. The inflection point is where the function changes concavity – for a basic cubic function, it is at the origin (0,0). For our transformed function \(y = x^3 - 2\), the inflection point shifts to \( (0, -2)\).

This symmetry means that for every point \( (x, y)\) on the graph, there is a corresponding point \( (-x, -y)\) reflected across the inflection point. Recognizing this can make it faster to plot points, as you can calculate the y-value for a positive x and then reflect it to find the corresponding point for its negative x counterpart. This method essentially doubles the number of points you have for graphing with half the work.

End Behavior of Functions

The end behavior of a function describes how the y-values behave as the x-values approach infinity or negative infinity. For cubic functions, as \(x\) approaches positive infinity, \(y\) also approaches positive infinity, similarly as \(x\) approaches negative infinity, \(y\) approaches negative infinity. This results in the cubic function's characteristic shape, which rises to the right and falls to the left.

In terms of \(y = x^3 - 2\), this means the left end of the graph will continue to decrease without bound, and the right end will increase without bound. This understanding of end behavior is essential when drawing the curve beyond the plotted points, ensuring the graph extends in the correct direction indefinitely. Keeping this in mind provides a clearer picture of the overall graph and guarantees the representation of the function's true nature.