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

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = -x^2 - 2x + 8\). This is in the standard form \(f(x) = ax^2 + bx + c\) where \(a = -1\), \(b = -2\), and \(c = 8\).
2. Recall that the vertex of a parabola defined by \(f(x) = ax^2 + bx + c\) has an \(x\)-coordinate given by the formula \(x = -\frac{b}{2a}\).
3. Substitute the values of \(a\) and \(b\) into the vertex formula: \(x = -\frac{-2}{2 \times -1}\).
4. Simplify the expression to find the \(x\)-coordinate of the vertex.
5. To find the \(y\)-coordinate of the vertex, substitute the \(x\)-value back into the original function: \(f(x) = -x^2 - 2x + 8\). Calculate \(f(\text{x-coordinate})\) to get the \(y\)-coordinate.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax^2 + 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) = ax^2 + bx + c, the vertex's x-coordinate is found using -b/(2a), and the y-coordinate is f(-b/(2a)).

Evaluating Functions

Evaluating a function means substituting a specific value for the variable and calculating the result. To find the vertex's y-coordinate, substitute the x-value of the vertex into the original quadratic function.