Question

Graph each function. Be sure to label three points on the graph. $$f(x)=x^{3}$$

Short answer

Plot points (-2, -8), (0, 0), and (2, 8) on the graph of f(x) = x^3 and connect them with a smooth curve.

Step-by-step solution

- Create a Table of Values

To begin graphing the function, create a table of values. Choose a few values for x, both positive and negative, and calculate the corresponding values of f(x) by plugging these x values into the function. For example, choose x values of -2, -1, 0, 1, and 2.

- Calculate y-values

Use the function f(x) = x^3 to find y-values for the chosen x values. For x = -2: f(-2) = (-2)^3 = -8For x = -1: f(-1) = (-1)^3 = -1For x = 0: f(0) = 0^3 = 0For x = 1: f(1) = 1^3 = 1For x = 2: f(2) = 2^3 = 8So the points are: (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8).

- Plot the Points

Plot the calculated points on a coordinate plane. Mark the points at (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8).

- Draw the Curve

Connect the points with a smooth curve to represent the function f(x) = x^3. Make sure the curve passes through all the plotted points and continues in both directions.

- Label Key Points

Label at least three of the plotted points on the graph to indicate their coordinates. For example, label the points (-2, -8), (0, 0), and (2, 8).

Key concepts

Table of Values

Creating a table of values is the first step in graphing any function, including cubic functions like \(f(x) = x^3\). To make a table of values, choose several x-values and compute the corresponding y-values using the function formula.
For instance, select x-values such as -2, -1, 0, 1, and 2.
Then, plug these x-values into \(f(x) = x^3\). For x = -2, \(f(-2) = (-2)^3 = -8\). For x = -1, \(f(-1) = (-1)^3 = -1\), and so on.
This process helps you gather a set of points that you will later plot on the graph.

Plotting Points

Once you have your table of values, you can start plotting the points on the graph. Each pair of (x, y) coordinates represents a point on the graph of the function.
For example, you have the following points from your table: (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8).
To plot a point, find the x-coordinate on the horizontal axis and the y-coordinate on the vertical axis. Place a dot where these coordinates intersect. This visual representation helps you see the shape of the function.

Coordinate Plane

The coordinate plane is a two-dimensional space where you plot your points. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis).
The point where these axes intersect is called the origin, marked as (0,0). Each point on the plane is defined by an (x, y) coordinate.
When plotting the points, ensure you are accurate with their placement. This accuracy affects the final shape of the graph. The points (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8) should be plotted correctly on the coordinate plane to reflect the function's behavior.

Smooth Curve

After plotting the points, connect them with a smooth curve to complete the graph of the cubic function. The goal is not to make straight lines between the points but to draw a flowing curve.
For \(f(x) = x^3\), the curve will pass smoothly through all the plotted points and extend indefinitely in both directions.
This smooth curve captures the continuous nature of the cubic function. When graphing, try to ensure the curve reflects key characteristics like local minima or maxima, inflection points, and the general rise and fall of the function.