Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)²+1

Verified step by step guidance

1. Identify the form of the quadratic function. The given function is in vertex form: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
2. Compare the given function \(f(x) = 2(x - 3)^2 + 1\) to the vertex form to find \(h\) and \(k\). Here, \(h = 3\) and \(k = 1\).
3. Recall that the vertex of the parabola is at the point \((h, k)\), so the vertex coordinates are \((3, 1)\).
4. Understand that since the coefficient \(a = 2\) is positive, the parabola opens upwards, confirming the vertex is a minimum point.
5. Summarize that the vertex coordinates for the parabola defined by \(f(x) = 2(x - 3)^2 + 1\) are \((3, 1)\).

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex directly without completing the square or using calculus.

Coordinates of the Vertex

The vertex of a parabola given in vertex form f(x) = a(x - h)^2 + k is the point (h, k). This point is either the maximum or minimum of the function depending on the sign of 'a'. For the function f(x) = 2(x - 3)^2 + 1, the vertex is at (3, 1).

Effect of the Coefficient 'a' on the Parabola

The coefficient 'a' in the quadratic function affects the parabola's width and direction. If 'a' is positive, the parabola opens upward, indicating a minimum vertex; if negative, it opens downward, indicating a maximum vertex. The larger the absolute value of 'a', the narrower the parabola.