Finding The Equation Of A Parabola With Vertex At (1 -6)

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#Introduction In the realm of quadratic functions and their graphical representations, parabolas, a critical task arises frequently: determining the equation of a parabola given specific information, such as its vertex. The vertex, the parabola’s extreme point (either the maximum or minimum), provides valuable insight into the parabola’s characteristics and equation. This article delves into the method of finding the equation of a parabola when its vertex is known, focusing on the case where the vertex is at the point (1, -6). We will explore how to leverage the vertex form of a quadratic equation to solve this type of problem effectively. This exploration will not only enhance your understanding of parabolas but also equip you with the skills to tackle related problems in mathematics and its applications.

Understanding the Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is a powerful tool for determining the equation of a parabola when its vertex is known. This form is expressed as:

y = a(x - h)^2 + k

Where:

  • (h, k) represents the coordinates of the vertex of the parabola.
  • a determines the direction and steepness of the parabola. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The magnitude of a affects how “stretched” or “compressed” the parabola is.

In our specific scenario, the vertex is given as (1, -6), which means h = 1 and k = -6. Substituting these values into the vertex form, we get:

y = a(x - 1)^2 - 6

To fully define the equation, we need to determine the value of a. This typically requires additional information, such as another point on the parabola or the parabola’s general shape. Without this additional information, we can express the equation in terms of a, highlighting the family of parabolas that share the same vertex. The vertex form allows us to quickly identify the vertex and understand how changes in a affect the parabola’s shape and orientation. Understanding the vertex form is crucial for both graphing parabolas and finding their equations, making it a fundamental concept in algebra and calculus.

Analyzing the Given Equations

In this section, we will analyze the provided quadratic equations to determine which one represents a parabola with a vertex at (1, -6). The equations we need to examine are:

  1. y = 3x^2 + 6x - 3
  2. y = 3x^2 - 6x - 3
  3. y = 3x^2 - 8x - 1
  4. y = 3x^2 - 3x - 6

To find the vertex of a parabola given in the standard form y = ax^2 + bx + c, we use the formula for the x-coordinate of the vertex:

h = -b / 2a

Then, we substitute this h value back into the equation to find the y-coordinate of the vertex, k.

Let’s apply this method to each equation:

Equation 1: y = 3x^2 + 6x - 3

  • a = 3, b = 6, c = -3
  • h = -6 / (2 * 3) = -1

Since the x-coordinate of the vertex is -1, this equation does not have a vertex at (1, -6).

Equation 2: y = 3x^2 - 6x - 3

  • a = 3, b = -6, c = -3
  • h = -(-6) / (2 * 3) = 1
  • Now, substitute x = 1 into the equation to find k:
    • y = 3(1)^2 - 6(1) - 3 = 3 - 6 - 3 = -6

Therefore, the vertex of this parabola is (1, -6), which matches the given vertex.

Equation 3: y = 3x^2 - 8x - 1

  • a = 3, b = -8, c = -1
  • h = -(-8) / (2 * 3) = 8 / 6 = 4 / 3

Since the x-coordinate of the vertex is 4/3, this equation does not have a vertex at (1, -6).

Equation 4: y = 3x^2 - 3x - 6

  • a = 3, b = -3, c = -6
  • h = -(-3) / (2 * 3) = 3 / 6 = 1 / 2

Since the x-coordinate of the vertex is 1/2, this equation does not have a vertex at (1, -6).

By analyzing each equation, we have determined that only Equation 2, y = 3x^2 - 6x - 3, represents a parabola with a vertex at (1, -6). This methodical approach of finding the vertex for each equation ensures accuracy and a clear understanding of why only one equation fits the given condition. The ability to quickly and accurately determine the vertex of a parabola is a crucial skill in algebra and calculus, enabling us to solve various problems related to quadratic functions.

Step-by-Step Solution Using the Vertex Formula

To definitively identify the equation representing a graph with a vertex at (1, -6), we can employ a step-by-step solution using the vertex formula. This method involves transforming the given quadratic equations into vertex form, which directly reveals the vertex coordinates. The vertex form of a quadratic equation is given by:

y = a(x - h)^2 + k

Where (h, k) is the vertex of the parabola. We will apply this to each of the provided equations.

Step 1: Reviewing the Vertex Formula

The vertex formula is derived from completing the square on the standard form of a quadratic equation, y = ax^2 + bx + c. The x-coordinate of the vertex, h, is given by:

h = -b / 2a

Once h is found, we substitute it back into the original equation to find the y-coordinate of the vertex, k. The vertex is then (h, k).

Step 2: Applying the Formula to Equation 1: y = 3x^2 + 6x - 3

  • Identify a = 3, b = 6, and c = -3.
  • Calculate h: h = -6 / (2 * 3) = -1.
  • Substitute h = -1 into the equation: y = 3(-1)^2 + 6(-1) - 3 = 3 - 6 - 3 = -6.
  • The vertex for this equation is (-1, -6), which does not match (1, -6).

Step 3: Applying the Formula to Equation 2: y = 3x^2 - 6x - 3

  • Identify a = 3, b = -6, and c = -3.
  • Calculate h: h = -(-6) / (2 * 3) = 1.
  • Substitute h = 1 into the equation: y = 3(1)^2 - 6(1) - 3 = 3 - 6 - 3 = -6.
  • The vertex for this equation is (1, -6), which matches the given vertex.

Step 4: Applying the Formula to Equation 3: y = 3x^2 - 8x - 1

  • Identify a = 3, b = -8, and c = -1.
  • Calculate h: h = -(-8) / (2 * 3) = 4 / 3.
  • Since h is not 1, this equation does not have a vertex at (1, -6).

Step 5: Applying the Formula to Equation 4: y = 3x^2 - 3x - 6

  • Identify a = 3, b = -3, and c = -6.
  • Calculate h: h = -(-3) / (2 * 3) = 1 / 2.
  • Since h is not 1, this equation does not have a vertex at (1, -6).

Step 6: Conclusion

After systematically applying the vertex formula to each equation, we can definitively conclude that the equation y = 3x^2 - 6x - 3 represents a graph with a vertex at (1, -6). This step-by-step approach not only provides the correct answer but also reinforces the understanding of how to find the vertex of a parabola given its equation. By transforming the quadratic equations and identifying their vertices, we ensure the accuracy of our solution and enhance our problem-solving skills in quadratic functions.

Conclusion

In conclusion, determining which quadratic equation represents a graph with a given vertex involves understanding and applying the vertex form of a quadratic equation or using the vertex formula. Through the detailed analysis of the provided equations, we identified that only y = 3x^2 - 6x - 3 has a vertex at (1, -6). This process underscores the importance of accurately calculating the vertex coordinates and comparing them with the given vertex. The step-by-step solution using the vertex formula provided a clear and methodical approach to solving the problem. Mastering these techniques is crucial for students and professionals in various fields, including mathematics, engineering, and computer graphics, where parabolas and quadratic functions are frequently encountered. This exploration not only answers the specific question but also enhances our ability to solve similar problems involving quadratic equations and their graphical representations.