Fill in the blank(s) to correctly complete each sentence. The vertex of the graph of ƒ(x) = x² + 2x + 4 has x-coordinate ____ .

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = x^2 + 2x + 4\).
2. Recall that the vertex of a parabola given by \(f(x) = ax^2 + bx + c\) has an x-coordinate found by the formula \(x = -\frac{b}{2a}\).
3. In this function, \(a = 1\) and \(b = 2\), so substitute these values into the formula: \(x = -\frac{2}{2 \times 1}\).
4. Simplify the expression to find the x-coordinate of the vertex: \(x = -\frac{2}{2}\).
5. Conclude that the x-coordinate of the vertex is the simplified value from the previous step.

Key Concepts

Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and properties of parabolas is essential for analyzing their vertices.

Vertex of a Parabola

The vertex of a parabola is the point where the graph changes direction, representing either a maximum or minimum value. For a quadratic function f(x) = ax^2 + bx + c, the vertex's x-coordinate can be found using the formula x = -b/(2a). This helps locate the exact position of the vertex on the x-axis.

Using the Vertex Formula

The vertex formula x = -b/(2a) is derived from completing the square or using calculus. It provides a quick way to find the x-coordinate of the vertex without graphing. Applying this formula to f(x) = x^2 + 2x + 4, where a=1 and b=2, yields the vertex's x-coordinate.