/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find the vertex for the parabola... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 23

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

Short Answer

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

Step by step solution

01

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.
02

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).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.鈥�

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.鈥�

Here鈥檚 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.

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Most popular questions from this chapter

Solve the formula for the specified variable. Because each variable is nonnegative, list only the principal square root. If possible, simplify radicals or eliminate radicals from denominators. $$A=p(1+r)^{2} \text { for } r$$

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