Question

Create a table for the reflection about the origin of the function
f(x) = (x + 2)} + 1.
Hint: First create a table of values for f (x) evaluated at x = 0, x = -1.,
and x = -2.

Answer 1

To reflect on the function \( f(x) = (x + 2) + 1 \), let's first simplify it:

\[
f(x) = x + 2 + 1 = x + 3
\]

Now, we can create a table of values by evaluating \( f(x) \) at \( x = 0, -1, \text{ and } -2 \).

Here is the table of values for \( f(x) \):

| \( x \) | \( f(x) = x + 3 \) | \( f(x) \) Value |
|:---------:|:-------------------:|:-----------------:|
| 0 | \( 0 + 3 \) | 3 |
| -1 | \( -1 + 3 \) | 2 |
| -2 | \( -2 + 3 \) | 1 |

### Table of Values:
| \( x \) | \( f(x) \) |
|:---------:|:-----------:|
| 0 | 3 |
| -1 | 2 |
| -2 | 1 |

### Reflection:
1. **Function's properties**: The function \( f(x) = x + 3 \) is a linear function, which means it has a constant rate of change and its graph is a straight line.
2. **Origin shift**: The function is a transformation of the basic linear function \( f(x) = x \) shifted up by 3 units.
3. **Values at specified points**: The function values show that as \( x \) decreases from 0 to -2, \( f(x) \) also decreases from 3 to 1. This indicates a downward trend in the function as \( x \) moves to the left on the x-axis.
4. **Graphical Representation**: Plotting these points will show a straight line with a positive slope, indicating that \( f(x) \) increases as \( x \) increases.

If you need any additional analyses or specific reflections about the function, feel free to ask!