Concept: Function Evaluation — substitute each $x$ into $f(x)=x^2$ to get $y$.

Step 1: Compute $y$ for each $x$.

$(-4)^2 = 16$, $(-3)^2 = 9$, $(-2)^2 = 4$, $(-1)^2 = 1$, $0^2 = 0$, $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$.

Step 2: Domain and range.

Domain is all real numbers $\mathbb{R}$. Range is $[0, \infty)$ because squares are never negative.

Step 3: Graph.

A parabola opening upward with vertex at $(0,0)$, symmetric about the $y$-axis.

Why this approach works

Function evaluation is the simplest operation in algebra: you replace the variable $x$ with the given number and compute. For $f(x)=x^2$, that means multiplying the input by itself. The table is just a list of these evaluations. Once you see the outputs, the domain (all real numbers) and range (all non‑negative reals) become obvious. The graph is a parabola — symmetric about the $y$-axis because $(-x)^2 = x^2$ — and its shape is determined by the points you’ve just computed.

1. Fill the table

For each $x$, compute $y = x^2$:

• $x = -4$: $y = (-4)^2 = 16$
• $x = -3$: $y = (-3)^2 = 9$
• $x = -2$: $y = (-2)^2 = 4$
• $x = -1$: $y = (-1)^2 = 1$
• $x = 0$: $y = 0^2 = 0$
• $x = 1$: $y = 1^2 = 1$
• $x = 2$: $y = 2^2 = 4$
• $x = 3$: $y = 3^2 = 9$
• $x = 4$: $y = 4^2 = 16$

The completed table:

2. Domain

The domain is the set of all inputs for which the function is defined. Since $f(x)=x^2$ works for every real number — you can square any real — the domain is all of $\mathbb{R}$.

3. Range

The range is the set of all possible outputs. A square is never negative: $x^2 \ge 0$ for every real $x$. Also, every non‑negative number is the square of some real number (its square root). So the range is all real numbers from $0$ upward.

4. Graph of $f(x) = x^2$

Plot the points from the table: $(-4,16)$, $(-3,9)$, $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$, $(4,16)$.

These points lie on a smooth curve called a parabola. The graph:

• Opens upward (since the coefficient of $x^2$ is positive).
• Has its vertex (lowest point) at $(0,0)$.
• Is symmetric about the $y$-axis.
• Gets steeper as $|x|$ increases.

A rough sketch:

y
|
16 -        *                 *
|
9  -    *                         *
|
4  - *                               *
|
1  -   *                           *
|
0  - *---*---*---*---*---*---*---*---*---> x
   -4  -3  -2  -1   0   1   2   3   4

(Imagine a smooth U‑shaped curve passing through all these points.)