Complete the table to show that h(x) is the reflection about the origin of the func...

1. Understand Reflection About the Origin: A function $h(x)$ is a reflection about the origin of a function $f(x)$ if for every point $(x, y)$ on the graph of $f(x)$, the point $(-x, -y)$ is on the graph of $h(x)$. This relationship can be expressed as $h(x) = -f(-x)$.

1. Analyze the Given Table for $f(x)$: The table for $f(x)$ provides the following points: $(0, 0)$, $(1, 1)$, and $(2, 8)$.

3. Determine the Corresponding Points for $h(x)$: Using the relationship $h(x) = -f(-x)$, we can find the values for $h(x)$ corresponding to the given $x$ values:
- For $x = 0$: $h(0) = -f(-0) = -f(0)$. Since $f(0) = 0$, $h(0) = -0 = 0$.
- For $x = 1$: $h(1) = -f(-1)$. We need to find $f(-1)$. From the given table, we only have positive $x$ values. However, if we assume $f(x)$ is an odd function (which is typical for reflections about the origin, e.g., $f(x) = x^3$), then $f(-1) = -f(1)$. Since $f(1) = 1$, $f(-1) = -1$. Therefore, $h(1) = -(-1) = 1$.
- For $x = 2$: $h(2) = -f(-2)$. Similarly, assuming $f(x)$ is an odd function, $f(-2) = -f(2)$. Since $f(2) = 8$, $f(-2) = -8$. Therefore, $h(2) = -(-8) = 8$.

4. Fill in the Table for $h(x)$: Based on the calculations above, the completed table for $h(x)$ is:
- When $x = 0$, $h(x) = 0$.
- When $x = 1$, $h(x) = 1$.
- When $x = 2$, $h(x) = 8$.

The table for $h(x)$ will have the following entries:
| x | h(x) |
|---|------|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |