Problem 62 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The \(x\) -coordinate of the vertex of the parabola whose equation is \(y=a x^{2}+b x+c\) is \(\frac{b}{2 a}\)

Step-by-Step Solution

Answer

The statement is false. The correct formula for the x-coordinate of the vertex of a parabola is \(-\frac{b}{2a}\).

Step 1: Understanding the Vertex of a Parabola

The vertex of a parabola given in the standard form y = ax^2 + bx + c is the maximum or minimum point of the parabola. For a parabola that opens upwards (a > 0), the vertex is the minimum point. For a parabola that opens downwards (a < 0), the vertex is the maximum point. The x-coordinate of the vertex can be found using the formula \(-\frac{b}{2a}\). As the statement provides \( \frac{b}{2a}\) as the x-coordinate of the vertex, it is incorrect.

Step 2: Correcting the Statement

To correct this false statement, the correct formula for the x-coordinate of the vertex of a given parabola in the standard form should be inserted. The correct formula for the x-coordinate of the vertex of a parabola is \(-\frac{b}{2a}\).

Key Concepts

Quadratic Equations

Quadratic equations are a cornerstone of algebraic studies and manifest as polynomial equations of the second degree. These equations typically take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The highest degree of the variable \(x\) is 2, which gives rise to the name "quadratic," derived from "quadratus," the Latin word for square.

Quadratic equations can be solved using various methods, such as factoring, completing the square, using the quadratic formula, or graphing. They play a vital role in modeling situations where relationships peak at a certain value and thus are crucial in fields like physics, engineering, and economics.

When graphed, a quadratic equation reveals a curve known as a parabola. Understanding how to identify the key features of a parabola, such as its vertex and axis of symmetry, is integral for interpreting quadratic functions effectively.

Standard Form of a Parabola

The standard form of a parabola provides a framework to easily identify and analyze its characteristics. Represented by the formula \(y = ax^2 + bx + c\), this form allows us to immediately recognize the direction in which the parabola opens and determine its vertex. In this equation:

• \(a\) dictates the width and direction of the parabola.
• \(b\) influences the position of the vertex along the x-axis.
• \(c\) represents the y-intercept, the point where the parabola intersects the y-axis.

To find the vertex of a parabola in standard form, we use the formula for the x-coordinate, which is \(-\frac{b}{2a}\). Once the x-coordinate is determined, we plug it back into the original equation to find the y-coordinate of the vertex. This gives us the precise point or "turning point" of the parabola. This efficient method assists in sketching the graph and understanding the parabola's behavior fully.

Parabola Opens Upwards or Downwards

Whether a parabola opens upwards or downwards is determined by the coefficient \(a\) in the quadratic equation \(y = ax^2 + bx + c\). This simple aspect can significantly change the interpretation and application of the equation.

• If \(a > 0\), the parabola opens upwards, resembling a U-shape, with its vertex representing the lowest point on the graph. This scenario occurs often when describing situations with minimum values, such as cost efficiencies or optimal solutions.
• If \(a < 0\), the parabola opens downwards, forming an inverted U-shape, where the vertex serves as the highest point, associated with maximum values. This is useful in scenarios such as finding maximum altitude or profit.

The direction a parabola opens not only affects its appearance but also influences the vertex's role as a maximum or minimum point, which plays a critical role in real-world problem-solving and decision-making.