Problem 23 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given. $$y=x^{2}+6 x$$

Step-by-Step Solution

Answer

The vertex of the parabola \(y = x^{2} + 6x\) is (-3, -9).

Step 1: Write the Equation in the Form of \(ax^{2}+bx+c\)

First, rewrite the equation \(y = x^{2}+6x\) as \(y = x^{2}+6x+0\). Here 0 is the c value which is not present in the original equation.

Step 2: Find the Coordinates of the Vertex

The vertex, (h, k), of a parabola, \(y = ax^{2} + bx + c\), is given by \(h = -b/(2a)\) and \(k = c - b^{2}/(4a)\). For the given equation, \(a = 1\), \(b = 6\), and \(c = 0\). Therefore, \(h = -6/(2*1) = -3\) and \(k = 0 - 6^{2}/(4*1) = -9\). Hence, the vertex of the parabola is (-3, -9).

Step 3: Verification

After finding the vertex, you will want to check its validity. To verify, rewrite the equation of the parabola in vertex form, \(y = a(x - h)^{2} + k\), which gives \(y = (x + 3)^{2} - 9\). By simplifying, we obtain \(y = x^{2} + 6x\), which is the original equation, hence the vertex (-3, -9) is correct.

Key Concepts

Quadratic Equation

Quadratic equations are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. In geometry, these equations describe parabolas, which are U-shaped curves. The coefficient \(a\) determines the direction of the parabola's opening. If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards.​

Here are a few essentials of quadratic equations:

• The highest exponent is 2, making it a second-degree polynomial.
• The graph of a quadratic equation is a parabola.
• Its general form is \(ax^2 + bx + c = 0\).

Understanding the properties of quadratic equations is crucial in solving problems related to parabolas in coordinate geometry.

Vertex Form

The vertex form of a quadratic equation, \(y = a(x - h)^2 + k\), is especially useful for identifying the vertex of a parabola. The vertex \((h, k)\) is a significant point on the curve that represents its maximum or minimum value, depending on whether the parabola opens upward or downward.​

Here’s how you can convert from standard form to vertex form:

• Complete the square on the quadratic expression \(ax^2 + bx + c\).
• Rewrite the equation in the form \(a(x - h)^2 + k\).
• The values of \(h\) and \(k\) are obtained from completing the square.

Let's consider the given equation \(y = x^2 + 6x + 0\). The process of completing the square allows us to find \(h = -3\) and \(k = -9\), leading to a vertex form of \(y = (x + 3)^2 - 9\). This verification shows that the vertex is accurately calculated.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe and analyze curves using coordinates on a plane. It provides a framework for representing geometric shapes algebraically.

• Axes: The coordinate plane is formed by two perpendicular lines called axes, labeled as the x-axis and y-axis.
• Points: Each point on the plane is represented by a pair of numbers \((x, y)\), known as coordinates.
• Graphs: Equations can be graphically represented, with each solution \((x, y)\) point forming part of the curve.

In the context of the given problem, we explore how the algebraic expression \(y = x^2 + 6x\) is graphically represented as a parabola. Analyzing such equations using coordinate geometry helps in understanding their properties and behavior, like identifying the vertex at \((-3, -9)\) to describe the curve's shape and direction.