Finding The Equation Of A Parabola With Vertex (1,-11) And Passing Through (4,64)
In the realm of mathematics, parabolas hold a significant place, appearing in various contexts from quadratic equations to the trajectory of projectiles. Understanding the properties and equations of parabolas is crucial for success in algebra, calculus, and beyond. This article delves into the process of determining the equation of a parabola given its vertex and another point it passes through. We will explore the standard form of a parabola, utilize the given information to solve for the unknown parameters, and arrive at the final equation. Let's embark on this mathematical journey together!
Understanding the Parabola and its Properties
Parabolas are U-shaped curves that are defined mathematically by a quadratic equation. They possess a unique set of properties that make them fascinating objects of study. The standard form of a parabola that opens upwards or downwards is given by:
f(x) = a(x - h)^2 + k
where:
- (h, k) represents the vertex of the parabola, which is the point where the parabola changes direction.
- 'a' determines the direction the parabola opens (upwards if a > 0, downwards if a < 0) and the width of the parabola. A larger absolute value of 'a' indicates a narrower parabola, while a smaller absolute value indicates a wider parabola.
The vertex is a critical point on the parabola. It represents the minimum value of the function if the parabola opens upwards (a > 0) and the maximum value if the parabola opens downwards (a < 0). The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
In this problem, we are given that the vertex of the parabola is (1, -11). This means that h = 1 and k = -11. We are also given that the parabola passes through the point (4, 64). This information will be crucial in determining the value of 'a', which will allow us to write the complete equation of the parabola.
Before we dive into the calculations, let's recap the key concepts. We know the standard form of a parabola, the significance of the vertex (h, k), and the role of the coefficient 'a'. By utilizing the given information and applying algebraic techniques, we will be able to find the equation of the parabola. So, let's proceed to the next section where we will use the given points to determine the equation.
Utilizing the Vertex and a Point to Find the Equation
To determine the equation of the parabola, we will substitute the given vertex and the point it passes through into the standard form of the parabola equation. We know that the vertex is (1, -11), so h = 1 and k = -11. Plugging these values into the standard form, we get:
f(x) = a(x - 1)^2 - 11
Now, we need to find the value of 'a'. We are given that the parabola passes through the point (4, 64). This means that when x = 4, f(x) = 64. We can substitute these values into the equation above:
64 = a(4 - 1)^2 - 11
Now, we have an equation with only one unknown variable, 'a'. We can solve for 'a' using basic algebraic operations. First, simplify the equation:
64 = a(3)^2 - 11
64 = 9a - 11
Next, add 11 to both sides of the equation:
64 + 11 = 9a
75 = 9a
Finally, divide both sides by 9 to isolate 'a':
a = 75 / 9
a = 25 / 3
So, we have found that a = 25/3. Now we have all the necessary components to write the equation of the parabola. We have the vertex (h, k) = (1, -11) and the value of 'a' = 25/3. Substituting these values into the standard form equation, we get:
f(x) = (25/3)(x - 1)^2 - 11
This is the equation of the parabola in vertex form. It represents a parabola with vertex (1, -11) that opens upwards (since a > 0) and passes through the point (4, 64). In the next section, we will compare our result with the given options to identify the correct answer and further validate our solution.
Comparing the Result with the Given Options
Having derived the equation of the parabola as f(x) = (25/3)(x - 1)^2 - 11, we now need to compare this with the options provided to identify the correct answer. The options are:
A. f(x) = -3x^2 + 6x - 14 B. f(x) = 3x^2 - 6x - 8 C. f(x) = 3(x + 1)^2 - 11 D. f(x) = (25/3)(x - 1)^2 - 11
By direct comparison, we can see that option D, f(x) = (25/3)(x - 1)^2 - 11, perfectly matches the equation we derived. This confirms our solution. However, let's briefly examine the other options to understand why they are incorrect.
Option A, f(x) = -3x^2 + 6x - 14, represents a parabola that opens downwards (since the coefficient of x^2 is negative), which contradicts the fact that our parabola opens upwards (a = 25/3 > 0). Therefore, option A is incorrect.
Option B, f(x) = 3x^2 - 6x - 8, represents a parabola that opens upwards. However, if we were to convert this equation to vertex form, we would find that its vertex is not (1, -11), making it an incorrect option.
Option C, f(x) = 3(x + 1)^2 - 11, is in vertex form, but its vertex is (-1, -11), which is different from the given vertex (1, -11). Hence, option C is also incorrect.
Through this process of comparison, we have confidently identified option D as the correct equation of the parabola. This exercise demonstrates the importance of understanding the standard form of a parabola, the significance of the vertex, and the ability to manipulate equations to arrive at the correct solution. In the final section, we will summarize the steps we took to solve this problem and highlight the key takeaways.
Summary and Key Takeaways
In this comprehensive guide, we have successfully determined the equation of a parabola given its vertex and a point it passes through. Let's recap the steps we followed and highlight the key takeaways from this problem.
- Understanding the Parabola and its Properties: We began by understanding the standard form of a parabola, f(x) = a(x - h)^2 + k, where (h, k) represents the vertex and 'a' determines the direction and width of the parabola. We emphasized the importance of the vertex as the point where the parabola changes direction and the role of 'a' in determining the parabola's shape.
- Utilizing the Vertex and a Point to Find the Equation: We substituted the given vertex (1, -11) into the standard form, resulting in the equation f(x) = a(x - 1)^2 - 11. Then, we used the fact that the parabola passes through the point (4, 64) to solve for 'a'. By substituting x = 4 and f(x) = 64 into the equation, we found that a = 25/3. This allowed us to write the complete equation of the parabola: f(x) = (25/3)(x - 1)^2 - 11.
- Comparing the Result with the Given Options: We compared the equation we derived with the provided options and confidently identified option D as the correct answer. We also analyzed the other options to understand why they were incorrect, reinforcing our understanding of the properties of parabolas.
Key Takeaways:
- The standard form of a parabola is a powerful tool for representing and analyzing parabolas.
- The vertex is a critical point that provides valuable information about the parabola's position and direction.
- Substituting given points into the equation allows us to solve for unknown parameters.
- Careful comparison and analysis are essential for verifying the solution.
By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of problems involving parabolas. Parabolas are fundamental in mathematics and have applications in various fields, so a solid understanding of their properties and equations is invaluable.
This step-by-step guide has provided you with the knowledge and skills to confidently find the equation of a parabola given its vertex and a point it passes through. Remember to practice these techniques to solidify your understanding and build your problem-solving abilities. With dedication and effort, you can excel in your mathematical endeavors!
