Finding The Equation Of A Parabola Turning Point (1, -2) And Passing Through (2, 1)

Understanding the Parabola and Its Properties

When exploring parabolas, it’s crucial to grasp their fundamental properties and equations. A parabola is a symmetrical U-shaped curve defined by a quadratic equation. The most common forms of the parabola equation are the standard form, y = ax² + bx + c, and the vertex form, y = a(x - h)² + k. The vertex form is particularly useful when the turning point (vertex) of the parabola is known, as it directly incorporates the coordinates of the vertex, which are represented as (h, k). Understanding these forms is the first step in determining the equation of a parabola given specific conditions. In this article, we will delve deep into how to find the equation of a parabola when its turning point and another point on the curve are provided. This involves substituting the known values into the vertex form of the parabola equation and solving for the unknown coefficient 'a'. The coefficient 'a' determines the direction and the steepness of the parabola. A positive 'a' indicates that the parabola opens upwards, while a negative 'a' indicates that it opens downwards. The absolute value of 'a' determines how stretched or compressed the parabola is; a larger absolute value means the parabola is narrower, and a smaller absolute value means it is wider. The turning point, also known as the vertex, is the point where the parabola changes direction. It is the minimum point if the parabola opens upwards and the maximum point if the parabola opens downwards. The ability to identify and use the vertex is essential for solving problems involving parabolas. The process of finding the equation of a parabola often involves algebraic manipulation and problem-solving skills. By mastering these techniques, you can confidently tackle various parabola related problems. This article aims to provide a comprehensive guide to this process, complete with detailed explanations and examples, ensuring a clear understanding of the concepts involved. Understanding the properties of parabolas, such as the vertex, axis of symmetry, and direction of opening, is crucial for solving problems related to quadratic functions. By using the vertex form of the equation, we can easily incorporate the turning point into our calculations, simplifying the process of finding the parabola's equation. This method is highly effective and widely used in various mathematical and scientific applications.

Setting Up the Vertex Form of the Equation

To effectively determine the parabola equation, especially given the turning point, initiating with the vertex form is the most strategic approach. The vertex form of a parabola equation is expressed as y = a(x - h)² + k, where (h, k) represents the coordinates of the turning point (vertex) and 'a' is a coefficient that dictates the parabola's shape and direction. Knowing the turning point's coordinates makes this form exceptionally advantageous because we can directly substitute these values into the equation. In our specific problem, the turning point is given as (1, -2). This means that h = 1 and k = -2. Substituting these values into the vertex form, we get y = a(x - 1)² - 2. This substitution simplifies the problem significantly, reducing the unknowns to just one variable, 'a'. The coefficient 'a' plays a crucial role in defining the parabola. It determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and also affects the parabola's width. A larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value results in a wider parabola. Therefore, finding the correct value of 'a' is essential to accurately define the parabola. The vertex form not only simplifies the initial setup but also provides a clear visual representation of the parabola's transformation from the basic y = x² parabola. The term (x - h)² represents a horizontal shift, and the term + k represents a vertical shift. This form allows us to easily see how the parabola is positioned on the coordinate plane relative to the origin. By using the vertex form, we are essentially building the parabola equation piece by piece. We start with the basic structure defined by the turning point and then refine it by determining the value of 'a'. This method is both intuitive and efficient, making it a preferred choice for solving parabola problems. The next step involves using the additional point provided to solve for 'a'. This will give us the complete equation of the parabola. The clarity and ease of use of the vertex form make it an indispensable tool for anyone working with parabolas. It allows for a systematic and organized approach to solving problems, ensuring accuracy and understanding.

Substituting the Given Point to Solve for 'a'

Having established the vertex form of the parabola equation as y = a(x - 1)² - 2, our next critical step involves determining the value of 'a'. To achieve this, we utilize the additional information provided: the parabola passes through the point (2, 1). This point gives us a specific x and y coordinate that must satisfy the parabola equation. By substituting x = 2 and y = 1 into the equation, we create an algebraic equation with 'a' as the only unknown variable. This equation can then be solved using standard algebraic techniques. Substituting the values, we get 1 = a(2 - 1)² - 2. Simplifying this equation is crucial. First, we evaluate the expression inside the parentheses: (2 - 1)² = 1² = 1. This simplifies the equation to 1 = a(1) - 2, which further simplifies to 1 = a - 2. Now, we have a simple linear equation in terms of 'a'. To isolate 'a', we add 2 to both sides of the equation: 1 + 2 = a - 2 + 2, which gives us 3 = a. Therefore, we have found that the value of 'a' is 3. This value is crucial because it completely defines the shape and direction of the parabola. A positive value of 'a' confirms that the parabola opens upwards, and the magnitude of 'a' indicates the steepness of the curve. The process of substituting a known point into the equation to solve for an unknown coefficient is a fundamental technique in algebra and calculus. It allows us to use specific information about a curve or function to determine its precise equation. This technique is not limited to parabolas; it can be applied to various types of functions and curves. By accurately substituting the given point and carefully simplifying the resulting equation, we can confidently find the value of 'a'. This step is essential in completing the equation of the parabola and ensuring that it accurately represents the given conditions. Once we have found 'a', we can substitute it back into the vertex form to obtain the final equation of the parabola. This completes the process of determining the parabola equation using the turning point and an additional point.

Finalizing the Equation and Verification

With the value of 'a' determined to be 3, the final step is to substitute this value back into the vertex form of the parabola equation, which we established earlier as y = a(x - 1)² - 2. Replacing 'a' with 3, we obtain the equation y = 3(x - 1)² - 2. This equation represents the parabola that has a turning point at (1, -2) and passes through the point (2, 1). To ensure the accuracy of our result, it is crucial to verify that this equation indeed satisfies the given conditions. Verification involves two key steps: confirming that the vertex is at (1, -2) and confirming that the point (2, 1) lies on the parabola. The vertex form of the equation directly indicates the vertex, which is (1, -2) in this case. This is evident from the terms (x - 1)² and -2, which correspond to the horizontal and vertical shifts from the origin, respectively. To verify that the point (2, 1) lies on the parabola, we substitute x = 2 into the equation and check if the resulting y-value is 1. Substituting x = 2 into y = 3(x - 1)² - 2, we get y = 3(2 - 1)² - 2. Simplifying this, we have y = 3(1)² - 2, which becomes y = 3 - 2, and finally, y = 1. This confirms that the point (2, 1) indeed lies on the parabola, as the calculated y-value matches the given y-coordinate. The completed equation, y = 3(x - 1)² - 2, can also be expanded into the standard form y = ax² + bx + c. Expanding the equation, we get y = 3(x² - 2x + 1) - 2, which simplifies to y = 3x² - 6x + 3 - 2, and further simplifies to y = 3x² - 6x + 1. This standard form provides an alternative representation of the parabola and can be useful for various analytical purposes. The process of finalizing and verifying the equation ensures that we have accurately determined the parabola that satisfies the given conditions. This methodical approach is essential for solving mathematical problems and building confidence in the results. The ability to find the equation of a parabola given specific conditions is a valuable skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics.

Conclusion

In conclusion, determining the equation of a parabola given a turning point and another point on the curve involves a systematic approach that utilizes the vertex form of the parabola equation. Starting with the vertex form, y = a(x - h)² + k, allows us to directly incorporate the coordinates of the turning point (h, k), simplifying the problem significantly. Substituting the turning point (1, -2) into the vertex form gives us y = a(x - 1)² - 2. The next crucial step is to use the additional point provided, (2, 1), to solve for the unknown coefficient 'a'. By substituting x = 2 and y = 1 into the equation, we obtain a linear equation in terms of 'a', which can be easily solved. In this case, we found that a = 3. Once the value of 'a' is determined, we substitute it back into the vertex form to obtain the final equation of the parabola: y = 3(x - 1)² - 2. To ensure the accuracy of our solution, we verify that the equation satisfies the given conditions. This involves confirming that the vertex is indeed at (1, -2) and that the point (2, 1) lies on the parabola. By substituting x = 2 into the equation, we verified that the resulting y-value is 1, confirming that the point (2, 1) is on the curve. The final equation, y = 3(x - 1)² - 2, represents the parabola that meets all the specified criteria. This process demonstrates the importance of understanding the properties of parabolas and the strategic use of different forms of the parabola equation. The vertex form is particularly useful when the turning point is known, as it simplifies the initial setup and allows for a clear understanding of the parabola's transformations. The ability to determine the equation of a parabola given specific conditions is a fundamental skill in mathematics with applications in various fields. By mastering this technique, you can confidently solve a wide range of problems involving quadratic functions and parabolas. The methodical approach outlined in this article provides a clear and effective strategy for tackling such problems, ensuring accuracy and a deep understanding of the concepts involved.