The given equation is $y = x^2 - 4$, and you are asked to graph it for integer values of $x$ from $-3$ to $3$, inclusive.

Let's first compute the corresponding $y$-values for each $x$-value in that range:

• For $x = -3$: $y = (-3)^2 - 4 = 9 - 4 = 5$
• For $x = -2$: $y = (-2)^2 - 4 = 4 - 4 = 0$
• For $x = -1$: $y = (-1)^2 - 4 = 1 - 4 = -3$
• For $x = 0$: $y = 0^2 - 4 = 0 - 4 = -4$
• For $x = 1$: $y = 1^2 - 4 = 1 - 4 = -3$
• For $x = 2$: $y = 2^2 - 4 = 4 - 4 = 0$
• For $x = 3$: $y = 3^2 - 4 = 9 - 4 = 5$

Now that we have the points $(-3, 5)$, $(-2, 0)$, $(-1, -3)$, $(0, -4)$, $(1, -3)$, $(2, 0)$, and $(3, 5)$, we can plot these on a graph. The shape of the graph will be a parabola, as this is a quadratic equation.

Would you like me to provide the graph as well?

Expanding the concept:

1. What is the general shape of a quadratic function like $y = x^2 - 4$?
2. How would the graph change if the equation were $y = -x^2 - 4$?
3. What is the vertex of the parabola for the equation $y = x^2 - 4$?
4. How does the value of $-4$ in $x^2 - 4$ affect the graph's position?
5. What happens to the graph if we change the equation to $y = (x - 1)^2 - 4$?

Tip:

In a quadratic function $y = ax^2 + bx + c$, the coefficient $a$ controls the direction and width of the parabola, while $c$ controls the vertical shift.