In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)^2+1

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Identify the standard form of a quadratic function, which is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Compare the given function \( f(x) = 2(x-3)^2 + 1 \) with the standard form to identify the values of \(a\), \(h\), and \(k\).

Notice that \(a = 2\), \(h = 3\), and \(k = 1\) from the function \( f(x) = 2(x-3)^2 + 1 \).

Recognize that the vertex \((h, k)\) of the parabola is \((3, 1)\).

Conclude that the coordinates of the vertex for the given quadratic function are \((3, 1)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the standard form and vertex form of quadratic functions is essential for analyzing their properties.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction, representing either the maximum or minimum value of the function. For a parabola in vertex form, f(x) = a(x-h)^2 + k, the vertex is located at the point (h, k). Identifying the vertex is crucial for graphing the parabola and understanding its behavior.

Completing the Square

Completing the square is a method used to transform a quadratic equation into vertex form, making it easier to identify the vertex. This technique involves manipulating the equation to create a perfect square trinomial, allowing for direct identification of the vertex coordinates. It is a fundamental skill in algebra that aids in solving quadratic equations and graphing parabolas.

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