To complete the table and graph for $y = x^2 - x - 3$, we calculate $y$ for each value of $x$:

Calculations:

For $x = -3$: $y = (-3)^2 - (-3) - 3 = 9 + 3 - 3 = 9$

For $x = -2$: $y = (-2)^2 - (-2) - 3 = 4 + 2 - 3 = 3$

For $x = -1$: $y = (-1)^2 - (-1) - 3 = 1 + 1 - 3 = -1$

For $x = 0$: $y = (0)^2 - (0) - 3 = 0 - 0 - 3 = -3$

For $x = 1$: $y = (1)^2 - (1) - 3 = 1 - 1 - 3 = -3$

For $x = 2$: $y = (2)^2 - (2) - 3 = 4 - 2 - 3 = -1$

For $x = 3$: $y = (3)^2 - (3) - 3 = 9 - 3 - 3 = 3$

Updated Table:

Graph:

1. Plot the points $(-3, 9)$, $(-2, 3)$, $(-1, -1)$, $(0, -3)$, $(1, -3)$, $(2, -1)$, and $(3, 3)$ on the grid.
2. Draw a smooth curve passing through the points.

Would you like me to create a visualization of the graph for you?

Related Questions:

1. How do you determine the vertex of the quadratic equation $y = x^2 - x - 3$?
2. What is the axis of symmetry for the graph of $y = x^2 - x - 3$?
3. How can you verify if the graph intersects the $x$-axis and calculate the roots?
4. How do you find the y-intercept for a quadratic equation?
5. What transformations can be applied to $y = x^2$ to arrive at $y = x^2 - x - 3$?

Tip:

Always verify your calculated points by substituting them back into the equation to ensure accuracy.