Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x²−8x+3

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = 2x^2 - 8x + 3\).
2. Recall that the vertex of a parabola defined by \(f(x) = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\).
3. Substitute the values of \(a = 2\) and \(b = -8\) into the formula: \(x = -\frac{-8}{2 \times 2}\).
4. Simplify the expression to find the x-coordinate of the vertex.
5. To find the y-coordinate of the vertex, substitute the x-coordinate back into the original function: \(f(x) = 2x^2 - 8x + 3\), and simplify.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax² + bx + c. Its graph is a parabola, which can open upwards or downwards depending on the sign of the coefficient a.

Vertex of a Parabola

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For f(x) = ax² + bx + c, the vertex's x-coordinate is found using -b/(2a), and the y-coordinate is f(-b/(2a)).

Completing the Square and Vertex Formula

Completing the square is a method to rewrite a quadratic function in vertex form, revealing the vertex coordinates directly. Alternatively, the vertex formula uses coefficients a and b to find the vertex without rewriting the function.