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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 5
Chapter 4, Problem 5

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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Key Concepts

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

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.
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Graphs of Logarithmic Functions

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.
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Horizontal Parabolas

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.
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Solving Quadratic Equations Using The Quadratic Formula
Related Practice
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Determine whether each statement is true or false. If false, explain why. The polynomial function ƒ(x)=2x5+3x4−8x3−5x+6ƒ(x)=2x^5+3x^4-8x^3-5x+6 has three variations in sign.

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