Question

Create a table for the reflection about the origin of the function \(f(x) =(x +2)^ 2 +1\).
Create a table of values for \(f\left(x\right)\) evaluated at \(x =0\), ., and \(x= -2\)

Answer 1

To create a table for the function \(f(x) = (x + 2)^2 + 1\), we first need to understand the function's properties, including its origin, which suggests the vertex form of a quadratic function.

### Origin of the Function
The function \(f(x) = (x + 2)^2 + 1\) is in vertex form \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

1. **Vertex**: In this function, the vertex is \((-2, 1)\).
2. **Direction**: Since the coefficient of the squared term \((x + 2)^2\) is positive (1), the parabola opens upwards.
3. **Y-Intercept**: The function will cross the y-axis when \(x = 0\).

### Table of Values
We will evaluate the function \(f(x)\) at several values including 0, -1, -2, and others for clarity.

\[
\begin{array}{|c|c|}
\hline
x & f(x) = (x + 2)^2 + 1 \\
\hline
-3 & f(-3) = (-3 + 2)^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2 \\
-2 & f(-2) = (-2 + 2)^2 + 1 = (0)^2 + 1 = 0 + 1 = 1 \\
-1 & f(-1) = (-1 + 2)^2 + 1 = (1)^2 + 1 = 1 + 1 = 2 \\
0 & f(0) = (0 + 2)^2 + 1 = (2)^2 + 1 = 4 + 1 = 5 \\
1 & f(1) = (1 + 2)^2 + 1 = (3)^2 + 1 = 9 + 1 = 10 \\
\hline
\end{array}
\]

### Summary of Values
- \(f(-3) = 2\)
- \(f(-2) = 1\)
- \(f(-1) = 2\)
- \(f(0) = 5\)
- \(f(1) = 10\)

This table clearly shows the values of the function \(f(x)\) at specified points. We can see that \(f(-2)\), the vertex, is the minimum value of the function, confirming its upward-opening parabola shape.