To analyze the reflection of the function \( f(x) = (x + 2)^2 + 1 \), we first need to create a table of values for \( f(x) \) evaluated at \( x = 0 \), \( x = -1 \), and \( x = -2 \).
Step 1: Calculate \( f(x) \) at the specified points
-
When \( x = 0 \): \[ f(0) = (0 + 2)^2 + 1 = 2^2 + 1 = 4 + 1 = 5 \]
-
When \( x = -1 \): \[ f(-1) = (-1 + 2)^2 + 1 = 1^2 + 1 = 1 + 1 = 2 \]
-
When \( x = -2 \): \[ f(-2) = (-2 + 2)^2 + 1 = 0^2 + 1 = 0 + 1 = 1 \]
Step 2: Create the table for \( -f(-x) \)
To find the values of \( -f(-x) \), we will evaluate the function \( f(x) \) at the corresponding negative values of \( x \):
- For \( x = 0 \), \( -f(-0) = -f(0) = -5 \)
- For \( x = 1 \), \( -f(-1) = -f(-1) = -2 \)
- For \( x = 2 \), \( -f(-2) = -f(-2) = -1 \)
Final Table
Here’s the completed table with values for \( f(x) \) and \( -f(-x) \):
| \( x \) | 0 | -1 | -2 | |---------|----|----|----| | \( f(x) \) | 5 | 2 | 1 | | \( -f(-x) \) | -5 | -2 | -1 |
This shows the values of \( f(x) \) at the specified points and the corresponding values of \( -f(-x) \), which can illustrate the reflection property of the function.