Question

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

Answer 1

To reflect on the function \( f(x) = (x+2)^2 + 1 \), we can analyze it in several key areas, such as its origin, transformations, vertex, and properties. Below is a structured table to summarize and reflect on these aspects.

| Aspect | Description |
|----------------------------|-------------------------------------------------------------|
| **Function** | \( f(x) = (x+2)^2 + 1 \) |
| **Type of Function** | Quadratic function (parabola) |
| **Standard Form** | The function is in vertex form: \( f(x) = a(x-h)^2 + k \) |
| **Vertex** | The vertex is at \( (-2, 1) \) (from \( h = -2 \) and \( k = 1 \)) |
| **Axis of Symmetry** | Vertical line \( x = -2 \) |
| **Direction of Opening** | Opens upwards (since the coefficient of the squared term is positive) |
| **Y-intercept** | Set \( x = 0 \): \( f(0) = (0 + 2)^2 + 1 = 4 + 1 = 5 \) (Y-intercept is \( (0, 5) \)) |
| **X-intercepts** | Set \( f(x) = 0 \): Solve \( (x + 2)^2 + 1 = 0 \); No real solutions (no X-intercepts) |
| **Transformation** | Shifted left 2 units and up 1 unit from the parent function \( f(x) = x^2 \) |
| **Reflection** | The function does not reflect across the x-axis; the vertex represents the minimum point. |
| **Range** | \( [1, \infty) \) (the function achieves a minimum value of 1) |

This table provides a comprehensive reflection on the function's properties, transformations, and appearance in the Cartesian plane.