Finding The Vertex Of Y = 2x² + 12x + 19 A Step-by-Step Guide

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In mathematics, quadratic equations represent parabolas, which are U-shaped curves. A key feature of a parabola is its vertex, which represents the minimum or maximum point of the curve. Understanding how to find the vertex is crucial in various applications, from optimizing functions to understanding the behavior of physical systems. This guide will provide a detailed explanation of how to find the vertex of the quadratic equation $y = 2x^2 + 12x + 19$, and we'll delve into the concepts and methods involved.

Understanding Quadratic Equations and Parabolas

Before we dive into the specifics of finding the vertex, let's establish a solid understanding of quadratic equations and their graphical representation, parabolas. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form: $y = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic equation is a parabola.

The parabola is a symmetrical curve. If 'a' is positive, the parabola opens upwards, indicating a minimum point. If 'a' is negative, the parabola opens downwards, indicating a maximum point. The vertex is this crucial turning point of the parabola. The vertex represents the minimum value of the function if the parabola opens upwards, and the maximum value if the parabola opens downwards. This makes the vertex a significant point for optimization problems.

Key characteristics of a parabola include:

  • Vertex: The turning point of the parabola, either the minimum or maximum point.
  • Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • Y-intercept: The point where the parabola intersects the y-axis (where x = 0).
  • X-intercept(s): The point(s) where the parabola intersects the x-axis (where y = 0). These are also known as the roots or zeros of the quadratic equation.

Methods to Find the Vertex

There are several methods to find the vertex of a parabola, each with its advantages. We will explore two primary methods:

  1. Using the Vertex Formula
  2. Completing the Square

1. Using the Vertex Formula

The vertex formula is a direct and efficient way to calculate the coordinates of the vertex. For a quadratic equation in the standard form $y = ax^2 + bx + c$, the vertex (h, k) can be found using the following formulas:

  • The x-coordinate (h) of the vertex is given by: $h = -b / 2a$
  • The y-coordinate (k) of the vertex is found by substituting the value of h back into the original equation: $k = f(h) = a(h)^2 + b(h) + c$

This formula is derived from the process of completing the square and provides a straightforward method for locating the vertex.

Applying the Vertex Formula to $y = 2x^2 + 12x + 19$

Let's apply the vertex formula to our equation $y = 2x^2 + 12x + 19$. First, we identify the coefficients:

  • a = 2
  • b = 12
  • c = 19

Now, we calculate the x-coordinate (h) of the vertex:

h=−b/2a=−12/(2∗2)=−12/4=−3h = -b / 2a = -12 / (2 * 2) = -12 / 4 = -3

Next, we substitute h = -3 back into the original equation to find the y-coordinate (k):

k=2(−3)2+12(−3)+19=2(9)−36+19=18−36+19=1k = 2(-3)^2 + 12(-3) + 19 = 2(9) - 36 + 19 = 18 - 36 + 19 = 1

Therefore, the vertex of the parabola is (-3, 1). This point represents the minimum value of the function since the coefficient 'a' (2) is positive, indicating that the parabola opens upwards.

2. Completing the Square

Completing the square is another powerful technique for finding the vertex of a parabola. This method involves rewriting the quadratic equation in vertex form, which directly reveals the coordinates of the vertex. The vertex form of a quadratic equation is:

y=a(x−h)2+ky = a(x - h)^2 + k

Where (h, k) is the vertex of the parabola. By completing the square, we transform the standard form of the quadratic equation into this vertex form.

Steps for Completing the Square

  1. Factor out 'a' from the $x^2$ and x terms:

    y=a(x2+(b/a)x)+cy = a(x^2 + (b/a)x) + c

  2. Complete the square inside the parentheses:

    • Take half of the coefficient of the x term (b/a), square it, and add it inside the parentheses. To maintain the equation's balance, subtract 'a' times this value outside the parentheses.

    • The value to add and subtract is $(b/2a)^2$

  3. Rewrite the expression inside the parentheses as a perfect square:

    y=a(x+b/2a)2+c−a(b/2a)2y = a(x + b/2a)^2 + c - a(b/2a)^2

  4. Simplify to obtain the vertex form:

    y=a(x−h)2+ky = a(x - h)^2 + k

    Where $h = -b/2a$ and $k = c - a(b/2a)^2$

Completing the Square for $y = 2x^2 + 12x + 19$

Let's apply the method of completing the square to our equation $y = 2x^2 + 12x + 19$.

  1. Factor out '2' from the $x^2$ and x terms:

    y=2(x2+6x)+19y = 2(x^2 + 6x) + 19

  2. Complete the square inside the parentheses:

    • Half of the coefficient of the x term (6) is 3, and squaring it gives 9. Add and subtract 9 inside the parentheses (and remember to multiply by 2 when subtracting outside).

    y=2(x2+6x+9−9)+19y = 2(x^2 + 6x + 9 - 9) + 19

  3. Rewrite the expression inside the parentheses as a perfect square:

    y=2((x+3)2−9)+19y = 2((x + 3)^2 - 9) + 19

  4. Distribute the '2' and simplify:

    y=2(x+3)2−18+19y = 2(x + 3)^2 - 18 + 19

    y=2(x+3)2+1y = 2(x + 3)^2 + 1

Now, the equation is in vertex form: $y = 2(x - (-3))^2 + 1$. Comparing this to the vertex form $y = a(x - h)^2 + k$, we can identify the vertex as (-3, 1).

Graphical Representation and Verification

Graphing the quadratic equation $y = 2x^2 + 12x + 19$ can visually confirm our calculated vertex. The graph is a parabola that opens upwards, and the vertex is the lowest point on the curve. Using graphing tools or software, you can plot the equation and observe that the vertex is indeed at the point (-3, 1).

This visual confirmation provides an additional layer of assurance that our calculations are correct. The symmetry of the parabola around the vertical line x = -3 (the axis of symmetry) is also evident from the graph, reinforcing the accuracy of our findings.

Practical Applications of Finding the Vertex

Finding the vertex of a quadratic equation is not just an academic exercise; it has numerous practical applications in various fields. Here are a few examples:

  1. Optimization Problems: The vertex represents the maximum or minimum value of a quadratic function. This is crucial in optimization problems, such as maximizing profit, minimizing costs, or finding the optimal trajectory of a projectile.

  2. Physics: In physics, projectile motion can be modeled using quadratic equations. The vertex of the parabola represents the maximum height reached by the projectile.

  3. Engineering: Engineers use quadratic equations to design structures, bridges, and other infrastructure. Finding the vertex helps determine the optimal dimensions and stability of these structures.

  4. Economics: Quadratic functions are used to model cost, revenue, and profit in economics. The vertex helps determine the break-even point and the maximum profit.

  5. Computer Graphics: Parabolas are used in computer graphics to create curves and shapes. The vertex is an important parameter in defining these curves.

Conclusion

In conclusion, finding the vertex of the quadratic equation $y = 2x^2 + 12x + 19$ involves understanding the properties of parabolas and applying appropriate methods. We have explored two primary methods: using the vertex formula and completing the square. Both methods lead to the same result, the vertex (-3, 1), which represents the minimum point of the parabola.

The vertex formula provides a direct calculation, while completing the square transforms the equation into vertex form, revealing the vertex coordinates. Graphing the equation visually confirms our results. Understanding how to find the vertex is essential for various applications, including optimization problems, physics, engineering, economics, and computer graphics.

By mastering these techniques, you can confidently analyze quadratic equations and parabolas, unlocking their potential in diverse fields. The ability to find the vertex is a fundamental skill in mathematics and its applications, providing valuable insights into the behavior of quadratic functions and their graphical representations.