Domain and Range of the Function

The function given is $f(x) = x^2 - 4$.

1. Domain: The domain of a function includes all possible values of $x$ for which the function is defined. Since $f(x) = x^2 - 4$ is a quadratic function (a polynomial of degree 2), it is defined for all real numbers.

   Therefore, the domain of $f(x)$ is: $\text{Domain} = (-\infty, \infty)$

2. Range: The range of a function is the set of all possible values of $f(x)$ as $x$ varies over the domain. For a quadratic function of the form $f(x) = ax^2 + bx + c$, if $a > 0$, the parabola opens upwards, and the minimum value occurs at the vertex. In this case, $a = 1$, so the parabola opens upwards.

   The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$. Here, $b = 0$, so: $x = 0$ Substituting $x = 0$ into the function: $f(0) = 0^2 - 4 = -4$ Thus, the minimum value of $f(x)$ is $-4$, and since the parabola opens upwards, the range is: $\text{Range} = [-4, \infty)$

Graphing the Function Using a Table of Ordered Pairs

To graph the function $f(x) = x^2 - 4$, we can create a table of values by choosing different values of $x$ and calculating the corresponding values of $f(x)$.

$x$	$f(x) = x^2 - 4$	Ordered Pair (x, f(x))
-3	$9 - 4 = 5$	(-3, 5)
-2	$4 - 4 = 0$	(-2, 0)
-1	$1 - 4 = -3$	(-1, -3)
0	$0 - 4 = -4$	(0, -4)
1	$1 - 4 = -3$	(1, -3)
2	$4 - 4 = 0$	(2, 0)
3	$9 - 4 = 5$	(3, 5)

Graph Interpretation

• The vertex of the parabola is at the point (0, -4).
• The parabola opens upwards.
• The points (-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), and (3, 5) are symmetric about the y-axis.

You can plot these points on a coordinate plane and connect them with a smooth curve to visualize the graph of the function.

Would you like more details or have any questions?

Related Questions:

1. What are the key features of the graph of a quadratic function?
2. How do you find the vertex of a parabola given its equation?
3. How does the value of "a" in a quadratic function $f(x) = ax^2 + bx + c$ affect the graph?
4. How can the graph of $f(x) = x^2 - 4$ be transformed into the graph of $g(x) = x^2 + 4$?
5. What are the roots of the function $f(x) = x^2 - 4$ and how do they relate to the graph?

Tip: When graphing a quadratic function, always find the vertex first, as it provides a useful reference point for plotting the rest of the graph.