which graph shows the axis of symmetry for the function f(x) = (x – 2)2 + 1?

Zank Er Valto

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11-11-2024

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Unlocking the Symmetry: Finding the Axis of Symmetry for f(x) = (x – 2)2 + 1

Understanding the axis of symmetry is crucial when analyzing quadratic functions. This line divides the parabola into two mirror images, helping us identify the vertex and understand the function's behavior. Let's delve into how to find the axis of symmetry for the function f(x) = (x – 2)2 + 1.

Understanding the Form: Vertex Form of Quadratic Functions

The given function, f(x) = (x – 2)2 + 1, is in vertex form, a helpful format for understanding the function's characteristics. This form is written as:

f(x) = a(x - h)2 + k

Where:

• a determines the direction of the parabola (upward if a > 0, downward if a < 0) and its stretch or compression.
• (h, k) represents the coordinates of the vertex.

The Key to Symmetry: The Vertex

The axis of symmetry always passes through the vertex of the parabola. In our function, f(x) = (x – 2)2 + 1, we can directly identify the vertex by comparing it to the vertex form:

• h = 2 and k = 1, so the vertex is located at (2, 1).

Defining the Axis of Symmetry: A Vertical Line

The axis of symmetry is a vertical line that passes through the vertex. Therefore, the axis of symmetry is the line:

x = 2

Visual Representation: The Graph

The graph of f(x) = (x – 2)2 + 1 will show a parabola opening upwards (since a = 1) with its vertex at (2, 1). The axis of symmetry will be a vertical line passing through the vertex, represented by the equation x = 2.

To summarize:

• The vertex form of a quadratic function helps identify the vertex, which is crucial for determining the axis of symmetry.
• The axis of symmetry is a vertical line that passes through the vertex.
• For the function f(x) = (x – 2)2 + 1, the axis of symmetry is the line x = 2.

By understanding the vertex form and its relationship to the axis of symmetry, you can easily identify this key feature of a quadratic function and analyze its behavior.