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

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Identify the quadratic function given: $f(x) = x^2 + 2x + 4$.

Recall that the vertex of a parabola given by $f(x) = ax^2 + bx + c$ has an x-coordinate found by the formula $x = -\frac{b}{2a}$.

In this function, $a = 1$ and $b = 2$, so substitute these values into the formula: $x = -\frac{2}{2 \times 1}$.

Simplify the expression to find the x-coordinate of the vertex: $x = -\frac{2}{2}$.

Conclude that the x-coordinate of the vertex is the simplified value from the previous step.

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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.

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