Graph F(x) = -x³ + 2 Domain, Range, And Graphing Guide

Understanding the Cubic Function

Before we proceed with the graphing, let's first understand the characteristics of a cubic function. A cubic function is a polynomial function of degree 3, generally expressed in the form f(x) = ax³ + bx² + cx + d, where 'a' is not equal to zero. The graph of a cubic function is a curve that can have up to two turning points, and its end behavior is such that as x approaches positive infinity, f(x) either approaches positive or negative infinity, and as x approaches negative infinity, f(x) behaves in the opposite manner.

The given function, f(x) = -x³ + 2, is a specific type of cubic function. The negative sign in front of the x³ term indicates that the graph will be reflected across the x-axis compared to the basic cubic function f(x) = x³. The '+ 2' term represents a vertical shift of the graph upwards by 2 units. These transformations significantly influence the shape and position of the graph, which we'll explore in detail as we plot the function.

Graphing the Function f(x) = -x³ + 2

To graph the function f(x) = -x³ + 2, we can follow a systematic approach that involves creating a table of values, plotting the points, and then connecting them to form the curve. This process will give us a visual representation of the function's behavior and help us understand its key features.

1. Creating a Table of Values

To create a table of values, we'll choose several x-values and calculate the corresponding f(x) values. Selecting a mix of positive, negative, and zero values will provide a comprehensive view of the function's graph. Here's a sample table:

2. Plotting the Points

Now, we'll plot the points from the table of values on a coordinate plane. Each point represents an (x, f(x)) pair, where x is the horizontal coordinate and f(x) is the vertical coordinate. Plotting these points will give us a visual scatter of the function's behavior.

The points we'll plot are: (-2, 10), (-1, 3), (0, 2), (1, 1), and (2, -6). These points provide a good starting point for understanding the curve of the function. More points can be plotted for increased accuracy, especially in regions where the curve changes direction rapidly.

3. Connecting the Points

After plotting the points, we'll connect them with a smooth curve. Remember that the graph of a cubic function is a continuous curve, so we shouldn't have any sharp corners or breaks in the graph. The curve should smoothly transition between the plotted points, reflecting the function's behavior.

As we connect the points, we'll notice the characteristic shape of a cubic function: it curves upwards on one side and downwards on the other. In this case, the negative coefficient of the x³ term causes the graph to decrease as x increases, and increase as x decreases. The vertical shift of +2 moves the entire graph upwards by two units.

Identifying the Domain and Range

Once we have the graph of the function, we can identify its domain and range. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (f(x) values) that the function can produce.

1. Domain

The domain of the cubic function f(x) = -x³ + 2 is all real numbers. This is because we can input any real number for x and obtain a corresponding real number output for f(x). There are no restrictions on the x-values we can use.

In interval notation, the domain is expressed as (-∞, ∞), indicating that it includes all numbers from negative infinity to positive infinity.

2. Range

The range of the cubic function f(x) = -x³ + 2 is also all real numbers. This is because the graph of the function extends infinitely upwards and downwards. As x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) approaches negative infinity. Therefore, the function can take on any real number value as output.

In interval notation, the range is expressed as (-∞, ∞), similar to the domain.

Summary of Domain and Range

• Domain: (-∞, ∞)
• Range: (-∞, ∞)

Conclusion

Graphing the function f(x) = -x³ + 2 provides a clear visual representation of its behavior. By creating a table of values, plotting the points, and connecting them with a smooth curve, we can understand the function's shape and identify its key features. Furthermore, by analyzing the graph, we can determine that the domain and range of the function are both all real numbers, expressed in interval notation as (-∞, ∞).

To graph the function f(x) = -x³ + 2, we create a table of values, plot the points, and connect them with a smooth curve. Use the graph to determine the domain and range of the function, expressing the answers in interval notation.

Graph f(x) = -x³ + 2: Domain, Range, and Graphing Guide