Determining Quadratic Function Parameters From Vertex And A Point

At the heart of understanding quadratic functions lies the vertex form, a powerful representation that unveils the key characteristics of these parabolic curves. The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where the parameters a, h, and k hold profound significance. This exploration delves into the intricacies of the vertex form, highlighting its importance in revealing the vertex, axis of symmetry, and the overall shape of the parabola. Understanding the vertex form of a quadratic equation is crucial for effectively solving various mathematical problems and real-world applications. The vertex form provides a clear picture of the parabola's key features, making it easier to analyze and manipulate quadratic functions.

Central to the vertex form is the vertex itself, represented by the coordinates (h, k). The vertex is the point where the parabola changes direction, marking either the minimum or maximum value of the function. The parameter h dictates the horizontal shift of the parabola, while k determines the vertical shift. A positive h shifts the parabola to the right, and a negative h shifts it to the left. Similarly, a positive k shifts the parabola upwards, and a negative k shifts it downwards. The vertex, therefore, serves as a reference point for understanding the parabola's position on the coordinate plane. This understanding is not only valuable in mathematics but also in various fields like physics and engineering, where parabolic trajectories are frequently encountered. For instance, the trajectory of a projectile under gravity can be modeled using a quadratic function, with the vertex representing the highest point reached by the projectile.

The parameter a plays a pivotal role in determining the shape and direction of the parabola. The magnitude of a dictates the parabola's width; a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider parabola. The sign of a determines the parabola's concavity. If a is positive, the parabola opens upwards, indicating a minimum value at the vertex. Conversely, if a is negative, the parabola opens downwards, indicating a maximum value at the vertex. The interplay between the parameters a, h, and k allows for a comprehensive understanding of the parabola's behavior. By manipulating these parameters, we can precisely control the parabola's position, shape, and direction. This control is essential in various applications, such as designing parabolic reflectors in satellite dishes or optimizing the shape of arches in architecture.

The axis of symmetry is another crucial feature readily identified from the vertex form. This vertical line passes through the vertex and divides the parabola into two symmetrical halves. The equation of the axis of symmetry is given by x = h, directly derived from the h-coordinate of the vertex. The axis of symmetry simplifies the process of sketching the parabola, as it provides a line of reflection for the curve. Knowing the axis of symmetry, we can easily plot points on one side of the parabola and then reflect them across the axis to obtain corresponding points on the other side. This symmetry is a fundamental property of parabolas and is exploited in numerous applications, including the design of optical lenses and the construction of suspension bridges.

In essence, the vertex form of a quadratic function provides a compact and informative representation that reveals the parabola's key characteristics. The vertex (h, k), the parameter a, and the axis of symmetry x = h collectively define the parabola's position, shape, and direction. Mastering the vertex form is essential for solving quadratic equations, graphing parabolas, and applying quadratic functions to real-world scenarios. This understanding forms a cornerstone of mathematical knowledge, enabling us to analyze and manipulate quadratic relationships effectively. The ability to convert between different forms of quadratic equations, such as standard form and vertex form, further enhances our problem-solving capabilities. Each form offers unique insights into the quadratic function, and the ability to switch between them allows us to tackle a wider range of problems.

The challenge presented involves finding the specific parameters a, h, and k for a quadratic function given certain information. We are told that the quadratic function has its vertex at the point (-7, 0), a crucial piece of information that directly reveals the values of h and k. Additionally, we know that the function passes through the point (-1, -3), which will allow us to determine the value of a. This process exemplifies how we can use given data points and the vertex form equation to define a unique quadratic function. The ability to determine these parameters is essential in various applications, such as curve fitting, where we aim to find a function that closely matches a set of data points.

The vertex coordinates directly provide the values for h and k. Since the vertex is at (-7, 0), we can immediately deduce that h = -7 and k = 0. This is a direct application of the definition of the vertex form, where the vertex coordinates are represented by (h, k). This initial step significantly simplifies the problem, as we now only need to determine the value of a. The clarity provided by the vertex form makes this identification straightforward, demonstrating the advantage of using this form when dealing with quadratic functions. The ability to quickly extract h and k from the vertex coordinates is a valuable skill in solving quadratic problems.

With h and k known, we can substitute these values into the vertex form equation: f(x) = a(x - (-7))^2 + 0, which simplifies to f(x) = a(x + 7)^2. This simplified equation now contains only one unknown, the parameter a. To find a, we utilize the fact that the function passes through the point (-1, -3). This means that when x = -1, the function value f(x) is equal to -3. Substituting these values into the equation gives us: -3 = a(-1 + 7)^2. This substitution transforms the problem into a simple algebraic equation that can be solved for a. The methodical approach of substituting known values into the equation is a fundamental technique in solving mathematical problems, particularly in algebra.

Now we solve the equation -3 = a(-1 + 7)^2 for a. First, simplify the expression inside the parentheses: -3 = a(6)^2. Then, square the 6: -3 = 36a. Finally, divide both sides by 36 to isolate a: a = -3/36, which simplifies to a = -1/12. This calculation provides the final piece of the puzzle, allowing us to completely define the quadratic function in vertex form. The process of solving for a demonstrates the power of algebraic manipulation in extracting information from equations.

With all parameters determined, we can now write the complete quadratic function in vertex form: f(x) = (-1/12)(x + 7)^2 + 0. This equation fully describes the quadratic function that satisfies the given conditions: a vertex at (-7, 0) and passing through the point (-1, -3). The negative value of a indicates that the parabola opens downwards, and the fraction -1/12 determines the parabola's width. This final equation showcases the power of the vertex form in concisely representing a quadratic function. The ability to construct the equation of a quadratic function from given information is a crucial skill in various mathematical and scientific contexts.

In summary, by carefully analyzing the given information and utilizing the properties of the vertex form of a quadratic function, we have successfully determined the parameters a, h, and k. The vertex at (-7, 0) immediately gave us h = -7 and k = 0. Using the point (-1, -3) and substituting it into the vertex form equation, we solved for a and found a = -1/12. This step-by-step approach demonstrates the logical progression of solving mathematical problems, starting from the given information and applying relevant formulas and techniques.

Therefore, the quadratic function in vertex form is f(x) = (-1/12)(x + 7)^2 + 0. This complete solution encapsulates the key concepts of vertex form and parameter determination. It highlights the importance of understanding the relationship between the parameters and the characteristics of the parabola. The ability to solve this type of problem is fundamental to mastering quadratic functions and their applications. This process not only reinforces mathematical skills but also develops problem-solving abilities that are valuable in various fields.

Detailed Breakdown of the Solution

To reiterate, we were given that a quadratic function has its vertex at (-7, 0) and passes through the point (-1, -3). Our goal was to find the values of a, h, and k in the vertex form equation f(x) = a(x - h)^2 + k. This type of problem often appears in algebra and precalculus courses, testing the understanding of quadratic functions and their properties. The systematic approach used here can be applied to similar problems involving different types of functions and equations.

1. Identify h and k: The vertex form equation directly provides the vertex coordinates (h, k). Since the vertex is given as (-7, 0), we have h = -7 and k = 0. This is the most straightforward step, demonstrating the direct connection between the vertex form and the vertex coordinates.

2. Substitute h and k: Substitute the values of h and k into the vertex form equation: f(x) = a(x - (-7))^2 + 0, which simplifies to f(x) = a(x + 7)^2. This substitution reduces the number of unknowns to just a, making the problem more manageable.

3. Use the point (-1, -3): Since the function passes through the point (-1, -3), we know that f(-1) = -3. Substitute x = -1 and f(x) = -3 into the equation: -3 = a(-1 + 7)^2. This substitution allows us to create an equation that can be solved for a.

4. Solve for a: Simplify the equation: -3 = a(6)^2, then -3 = 36a. Divide both sides by 36 to get a = -3/36, which simplifies to a = -1/12. This algebraic manipulation is a crucial step in finding the value of a.

5. Write the final equation: Substitute the values of a, h, and k into the vertex form equation: f(x) = (-1/12)(x + 7)^2 + 0. This is the complete quadratic function in vertex form, satisfying the given conditions.

Importance of Understanding Vertex Form

Understanding the vertex form of a quadratic function is not just about solving specific problems; it's about developing a deeper understanding of quadratic functions and their applications. The vertex form provides valuable insights into the parabola's characteristics, such as its vertex, axis of symmetry, and direction of opening. This understanding is crucial in various fields, including physics, engineering, and economics.

In physics, quadratic functions are used to model projectile motion, where the vertex represents the maximum height reached by the projectile. In engineering, quadratic functions are used in the design of parabolic reflectors, such as those used in satellite dishes and solar concentrators. In economics, quadratic functions can be used to model cost and revenue curves, helping businesses optimize their operations.

By mastering the vertex form and the techniques for determining its parameters, you gain a powerful tool for analyzing and solving problems involving quadratic functions. This skill will be invaluable in your further studies in mathematics and related fields. The problem-solving process outlined here can be applied to a wide range of mathematical problems, fostering critical thinking and analytical skills.

In conclusion, we have successfully determined the parameters a, h, and k for the given quadratic function using the vertex form and the provided information. The vertex form, f(x) = a(x - h)^2 + k, is a powerful tool for understanding and manipulating quadratic functions. By identifying the vertex and using a given point, we were able to find the specific equation of the quadratic function. This process demonstrates the importance of understanding the relationship between the parameters and the graphical representation of the function. The final equation, f(x) = (-1/12)(x + 7)^2 + 0, represents the unique quadratic function that satisfies the given conditions. This exercise not only reinforces mathematical skills but also highlights the practical applications of quadratic functions in various fields.