The equation given is $y = x^3 - 1$. To interpret this:

1. Equation Analysis: This equation describes a cubic function. For each value of $x$, the output $y$ is determined by cubing $x$ and subtracting 1.

2. Table of Values: The table provided shows the following pairs of values:

   x & y \\ \hline 2 & 7 \\ 1 & 0 \\ 0 & -1 \\ -1 & -2 \\ -2 & -9 \\ \end{array}$$ These values are correct based on the equation: $$y = x^3 - 1$$ For example: - For $$x = 2$$: $$y = 2^3 - 1 = 8 - 1 = 7$$ - For $$x = 1$$: $$y = 1^3 - 1 = 1 - 1 = 0$$ - For $$x = 0$$: $$y = 0^3 - 1 = 0 - 1 = -1$$ - For $$x = -1$$: $$y = (-1)^3 - 1 = -1 - 1 = -2$$ - For $$x = -2$$: $$y = (-2)^3 - 1 = -8 - 1 = -9$$

3. Graphing the Function: The blank grid on the left can be used to plot the points $(x, y)$ from the table: $(2, 7), (1, 0), (0, -1), (-1, -2), (-2, -9)$ You would mark these points on the grid and connect them to show the cubic curve.

4. Is this a function?: Yes, this is a function. Each value of $x$ corresponds to exactly one value of $y$, meaning it passes the vertical line test (a vertical line would intersect the graph at only one point).

Would you like a more detailed explanation or help with graphing?

Additional Questions:

1. What is the general shape of a cubic function on a graph?
2. How does the vertical line test work to determine if something is a function?
3. Can a function have the same output for different inputs?
4. How would the graph of $y = x^3 - 1$ differ from $y = x^3$?
5. What is the significance of the constant term $-1$ in the equation $y = x^3 - 1$?

Tip:

When graphing cubic functions, note that they usually have an "S" shape, with one end going to positive infinity and the other to negative infinity.