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

Verified step by step guidance

1. Step 1: Identify the equation given, which is \( y = x^3 - 1 \). This is a cubic function.
2. Step 2: Create a table of values for \( x \) using the given values: \(-3, -2, -1, 0, 1, 2, 3\).
3. Step 3: For each \( x \) value, substitute it into the equation \( y = x^3 - 1 \) to find the corresponding \( y \) value.
4. Step 4: Plot each \( (x, y) \) pair on a coordinate plane to visualize the graph of the function.
5. Step 5: Connect the plotted points smoothly, considering the nature of a cubic function, which typically has an 'S' shape curve.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the independent variable (x) and the dependent variable (y). For the equation y = x^3 - 1, you will calculate y for each given x value, creating a set of points that can be connected to form the graph. Understanding how to interpret and create graphs is essential for analyzing the behavior of functions.

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form y = ax^3 + bx^2 + cx + d. The function y = x^3 - 1 is a specific cubic function where the leading coefficient is 1 and the constant term is -1. Cubic functions exhibit unique characteristics, such as having one or two turning points and can model various real-world scenarios.

Evaluating Functions

Evaluating functions involves substituting specific values of the independent variable into the function to find the corresponding output. In this case, you will substitute x values of -3, -2, -1, 0, 1, 2, and 3 into the equation y = x^3 - 1 to determine the corresponding y values. This process is fundamental for graphing and understanding the function's behavior.