Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = x³

Verified step by step guidance

1. Identify the given equation: y = x^3. This is a cubic function, which means the graph will have a characteristic S-shape, passing through the origin (0, 0).
2. Create a table of values for the given x-values: x = -3, -2, -1, 0, 1, 2, 3. For each x-value, substitute it into the equation y = x^3 to calculate the corresponding y-value. For example, when x = -3, y = (-3)^3 = -27.
3. Complete the table of values by calculating y for all the given x-values. The table will look like this: x = -3, -2, -1, 0, 1, 2, 3 and corresponding y-values will be calculated as y = x^3.
4. Plot the points from the table of values on a coordinate plane. For example, plot (-3, -27), (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8), and (3, 27).
5. Draw a smooth curve through the plotted points to represent the graph of the cubic function y = x^3. Ensure the curve reflects the S-shape characteristic of cubic functions, with the graph decreasing for negative x-values, passing through the origin, and increasing for positive x-values.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values). For the equation y = x^3, each x-value corresponds to a specific y-value calculated by cubing x. Understanding how to plot these points accurately is essential for interpreting the function's behavior.

Cubic Functions

Cubic functions are polynomial functions of degree three, characterized by their general form y = ax^3 + bx^2 + cx + d. The function y = x^3 is a simple cubic function where a = 1, b = 0, c = 0, and d = 0. These functions typically exhibit an S-shaped curve and can have one or two turning points, influencing their graph's shape and direction.

Evaluating Functions

Evaluating functions involves substituting specific values for the variable to find corresponding outputs. In this case, substituting x = -3, -2, -1, 0, 1, 2, and 3 into the equation y = x^3 allows us to calculate the y-values. This process is crucial for generating the points needed to accurately graph the function.