Finding The Value Of B Given The Axis Of Symmetry In A Quadratic Function

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In mathematics, understanding the properties of quadratic functions is crucial for solving a variety of problems. One such property is the axis of symmetry, which provides valuable information about the graph of the function. This article delves into the process of determining the value of the coefficient b in a quadratic function, given the axis of symmetry. We will specifically address the problem where the function is given as f(x) = (1/4)x^2 + bx + 10, and the axis of symmetry is x = 6. This exploration will not only help in solving this particular problem but also enhance the understanding of quadratic functions and their characteristics.

Understanding Quadratic Functions and the Axis of Symmetry

To effectively solve this problem, it's essential to have a solid grasp of quadratic functions and the axis of symmetry. A quadratic function is a polynomial function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the vertex of the parabola, which is the point where the parabola changes direction. The x-coordinate of the vertex, and thus the equation of the axis of symmetry, can be found using the formula x = -b / 2a.

In our case, the given quadratic function is f(x) = (1/4)x^2 + bx + 10. Here, a = 1/4, b is the unknown we need to find, and c = 10. The axis of symmetry is given as x = 6. By understanding the relationship between the coefficients of the quadratic function and the axis of symmetry, we can set up an equation and solve for b. This involves substituting the known values into the axis of symmetry formula and solving for the unknown variable. The process highlights the importance of recognizing key features of quadratic functions and using relevant formulas to determine unknown parameters.

Problem Statement: Finding b in f(x) = (1/4)x^2 + bx + 10 Given the Axis of Symmetry x = 6

Let's restate the problem clearly. We are given the quadratic function f(x) = (1/4)x^2 + bx + 10, and we know that the axis of symmetry for the graph of this function is x = 6. The task is to find the value of the coefficient b. This problem requires us to utilize the formula for the axis of symmetry of a quadratic function and apply it to the specific parameters provided. By setting up the equation correctly and solving for b, we can determine the precise value that satisfies the given condition. This exercise not only reinforces the understanding of quadratic functions but also demonstrates the practical application of mathematical formulas in problem-solving scenarios.

Solution

To find the value of b, we will use the formula for the axis of symmetry, which is given by x = -b / 2a. In this problem, we have a = 1/4 and the axis of symmetry is x = 6. We can substitute these values into the formula and solve for b. The step-by-step solution is outlined below:

  1. Write down the formula for the axis of symmetry: x = -b / 2a
  2. Substitute the given values: We know that x = 6 and a = 1/4, so we substitute these into the formula: 6 = -b / (2 * (1/4))
  3. Simplify the equation: Simplify the denominator: 6 = -b / (1/2)
  4. Multiply both sides by (1/2): To isolate the term with b, multiply both sides of the equation by 1/2: 6 * (1/2) = -b 3 = -b
  5. Solve for b: Multiply both sides by -1 to solve for b: b = -3

Therefore, the value of b is -3. This result means that the quadratic function is f(x) = (1/4)x^2 - 3x + 10, and its graph has an axis of symmetry at x = 6. This process demonstrates how understanding and applying the formula for the axis of symmetry can help in determining unknown coefficients in quadratic functions. The algebraic manipulation and step-by-step approach are crucial for arriving at the correct solution.

Verifying the Solution

To ensure that our solution is correct, we can substitute the value of b back into the original equation and check if the axis of symmetry is indeed x = 6. We found that b = -3, so the quadratic function becomes f(x) = (1/4)x^2 - 3x + 10. To verify, we use the axis of symmetry formula x = -b / 2a with the new value of b. Substituting a = 1/4 and b = -3 into the formula, we get:

x = -(-3) / (2 * (1/4)) x = 3 / (1/2) x = 3 * 2 x = 6

This confirms that the axis of symmetry is indeed x = 6, which matches the given information in the problem. This verification step is essential in mathematics to avoid errors and ensure the accuracy of the solution. It also reinforces the understanding of the relationship between the coefficients of a quadratic function and its axis of symmetry. By verifying the solution, we gain confidence in our answer and the process we followed.

Alternative Approaches to Solving the Problem

While the standard method of using the axis of symmetry formula is direct and efficient, there are alternative approaches to solving this problem that can provide a deeper understanding of quadratic functions. One such method involves completing the square. By rewriting the quadratic function in vertex form, we can directly identify the vertex of the parabola, and hence the axis of symmetry. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, and x = h is the axis of symmetry.

Another approach involves using the properties of the roots of the quadratic equation. The axis of symmetry lies exactly in the middle of the two roots (if they exist). If we can find the roots of the quadratic equation (1/4)x^2 + bx + 10 = 0 in terms of b, we can then find the midpoint of these roots and set it equal to 6, thereby solving for b. This method, however, may be more complex and time-consuming compared to the direct application of the axis of symmetry formula. Nevertheless, exploring these alternative methods can enhance problem-solving skills and provide a more comprehensive understanding of quadratic functions.

Common Mistakes to Avoid

When solving problems involving quadratic functions and the axis of symmetry, there are several common mistakes that students often make. One frequent error is misinterpreting the formula for the axis of symmetry. The correct formula is x = -b / 2a, but students sometimes forget the negative sign or mix up the coefficients. It is crucial to memorize the formula accurately and apply it correctly.

Another common mistake is in the algebraic manipulation of the equation. When substituting values and solving for b, it's essential to follow the order of operations and avoid errors in arithmetic. For example, when dealing with fractions, it's important to simplify correctly and ensure that all operations are performed accurately. Additionally, students may make mistakes when completing the square or finding the roots of the quadratic equation, which are alternative methods for solving this type of problem. To avoid these mistakes, it's essential to practice solving similar problems and carefully review each step of the solution process.

Conclusion

In summary, we have successfully determined the value of b in the quadratic function f(x) = (1/4)x^2 + bx + 10, given that the axis of symmetry is x = 6. By applying the formula x = -b / 2a, we found that b = -3. We also verified our solution by substituting the value of b back into the equation and confirming that the axis of symmetry is indeed x = 6. This problem highlights the importance of understanding the relationship between the coefficients of a quadratic function and its graphical properties, particularly the axis of symmetry. Furthermore, we explored alternative approaches to solving the problem and discussed common mistakes to avoid.

Mastering quadratic functions and their properties is essential for success in mathematics. This exercise not only reinforces the application of specific formulas but also enhances problem-solving skills and the ability to approach mathematical challenges with confidence. By understanding the concepts and practicing regularly, students can develop a strong foundation in quadratic functions and excel in related topics.