Question

Create a table for the reflection about the origin of the function f(x)=(x+2)2+1
𝑓
(
𝑥
)
=
(
𝑥
+
2
)
2
+
1
.

Hint: First create a table of values for f(x)
𝑓
(
𝑥
)
evaluated at x=0
𝑥
=
0
, x=−1

Answer 1

To create a table of values for the function \( f(x) = (x + 2)^2 + 1 \), we'll evaluate the function at the specified values of \( x \) (i.e., \( x = 0 \) and \( x = -1 \)) and include additional values for a more complete reflection.

### Step 1: Calculate the values of \( f(x) \)

1. **When \( x = -2 \)**:
\[
f(-2) = (-2 + 2)^2 + 1 = 0^2 + 1 = 1
\]

2. **When \( x = -1 \)**:
\[
f(-1) = (-1 + 2)^2 + 1 = 1^2 + 1 = 1 + 1 = 2
\]

3. **When \( x = 0 \)**:
\[
f(0) = (0 + 2)^2 + 1 = 2^2 + 1 = 4 + 1 = 5
\]

4. **When \( x = 1 \)**:
\[
f(1) = (1 + 2)^2 + 1 = 3^2 + 1 = 9 + 1 = 10
\]

5. **When \( x = -3 \)**:
\[
f(-3) = (-3 + 2)^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2
\]

6. **When \( x = -4 \)**:
\[
f(-4) = (-4 + 2)^2 + 1 = (-2)^2 + 1 = 4 + 1 = 5
\]

### Step 2: Create the table of values

| \( x \) | \( f(x) \) |
|----------|------------|
| -4 | 5 |
| -3 | 2 |
| -2 | 1 |
| -1 | 2 |
| 0 | 5 |
| 1 | 10 |

### Reflection about the function \( f(x) \)

The function \( f(x) = (x + 2)^2 + 1 \) is a quadratic function that opens upward and has its vertex at the point \((-2, 1)\).

- **Vertex**: The vertex of the function is clearly at the lowest point of the parabola since it opens upward. Thus, \( (-2, 1) \) represents the minimum value of \( f(x) \).

- **Symmetry**: The function is symmetric about the line \( x = -2 \), which is the vertical line that passes through the vertex.

- **Behavior**: As \( x \) moves away from -2, either towards \( -3 \) or \( -1 \), the value of \( f(x) \) increases. This can be observed in the table.

- **Reflection**: The values of \( f(x) \) for \( -3 \) and \( -1 \) are the same (\( f(-3) = 2 \) and \( f(-1) = 2 \)), demonstrating the reflective property of quadratic functions.

In conclusion, the function \( f(x) = (x + 2)^2 + 1 \) has its origin at the vertex and reflects symmetrically about the vertical line \( x = -2 \).