Question

Use a graphing utility to graph \(f\) for \(c=-2,0\), and 2 in the same viewing window. (a) \(f(x)=x^{3}+c\) (b) \(f(x)=(x-c)^{3}\) (c) \(f(x)=(x-2)^{3}+c\) In each case, compare the graph with the graph of \(y=x^{3}\).

Short answer

Graphs of \(f(x) = x^{3} + c\) vary in vertical position according to the value of \(c\). Graphs of \(f(x) = (x - c)^{3}\) show horizontal shifts of the graph in relation to changes in \(c\). \(f(x) = (x - 2)^{3} + c\) indicates that a horizontal shift of 2 units to the right from the origin combined with a vertical shift equal to \(c\) occurs.

Step-by-step solution

Graph \(f(x) = x^{3} + c\) with a graphing utility for \(c=-2,0,2\)

For \(c=-2\), the function to graph is \(f(x) = x^{3} - 2\). For \(c=0\), the function is identical to \(f(x) = x^{3}\). For \(c=2\), the function to graph is \(f(x) = x^{3} + 2\). Then compare each graph with that of \(y = x^{3}\).

Graph \(f(x) = (x - c)^{3}\) with a graphing utility for \(c=-2,0,2\)

For \(c=-2\), the function to graph is \(f(x) = (x + 2)^{3}\). For \(c=0\), again, the function is identical to \(f(x) = x^{3}\). For \(c=2\), the function to graph is \(f(x) = (x - 2)^{3}\). Look for shifts of the basic graph \((x-c)^3\) compared with \(x^3\).

Graph \(f(x) = (x - 2)^{3} + c\) with a graphing utility for \(c=-2,0,2\) and comparison

For \(c=-2\), the function to graph is \(f(x) = (x - 2)^{3} - 2\). For \(c=0\), here \(f(x) = (x - 2)^{3}\). And for \(c=2\), the function to graph is \(f(x) = (x - 2)^{3} + 2\). Compare each with the graph of \(y = x^{3}\) to see how horizontal shifts and vertical shifts simultaneously affect the equation \(y = x^{3}\).

Key concepts

Function Transformations

Function transformations involve changing the position or shape of a graph. In our exercise, we are dealing with both vertical and horizontal transformations, illustrated in functions like \(f(x) = x^3 + c\) and \(f(x) = (x - c)^3\).

• **Vertical Shifts**: This happens when a constant \(c\) is added or subtracted from the entire function. For example, in \(f(x) = x^3 + c\), the graph shifts vertically. When \(c > 0\), the graph shifts upward; when \(c < 0\), it shifts downward.

• **Horizontal Shifts**: Occur when the constant \(c\) is subtracted within the function’s argument, as seen in \(f(x) = (x - c)^3\). Here, the graph shifts horizontally. If \(c > 0\), the graph shifts to the right, while for \(c < 0\), it shifts to the left.

These adjustments allow you to manipulate and understand graphs dynamically by observing how changes in \(c\) affect the graph’s position.

Graphing Utilities

Graphing utilities are tools that help visualize mathematical functions. They can be either physical devices like graphing calculators or digital applications on computers and mobile devices.

• **Ease of Visualization**: These tools let you input the mathematical equation and instantly get a visual image of the graph. This is especially helpful in analyzing complex transformations.

• **Comparison Capabilities**: Graphing utilities allow multiple graphs to be displayed in the same viewing window, making it easy to compare different functions. Like in the exercise, you can use these tools to plot \(f(x) = x^3 + c\) for various values of \(c\) simultaneously.

• **Analysis Features**: Besides graphing, many utilities offer additional analysis features such as finding intercepts, calculating derivatives, or evaluating integrals. These features extend the capability of what students can learn from a single function.

Ultimately, graphing utilities are essential for modern mathematics, making abstract concepts more tangible and understandable.

Comparing Graphs

Comparing graphs is a crucial skill in understanding the effects of transformations. By observing differences and similarities between graphs, you can draw significant conclusions about their behavior and transformations.

• **Visual Shifts**: In functions like \(f(x) = x^3 + c\) and \(f(x) = (x - c)^3\), compare their graphs to \(y = x^3\). Notice how changes in the constant \(c\) alter the position. For instance, a positive \(c\) in \(x^3 + c\) will elevate the curve, whereas \((x - c)^3\) slides the graph left or right.

• **Simultaneous Comparisons**: Placing multiple transformations in one graph highlights differences in shifts. This visibility is particularly useful in exercises dealing with simultaneous vertical and horizontal shifts, like seen in \((x - 2)^3 + c\).

• **Insights into General Patterns**: By comparing transformations, patterns across different functions become clear. Understand how each type of shift affects the graph, helping predict the outcomes of other transformations without needing to graph them multiple times each.

Through effective comparison, you enhance deeper comprehension of function behaviors and transformations, foundational in many areas of mathematics.