Finding The Value Of B Given Axis Of Symmetry In Quadratic Function
In mathematics, quadratic functions play a pivotal role, and understanding their properties is crucial for various applications. One essential property of a quadratic function is its axis of symmetry, which provides valuable information about the function's graph and behavior. This article delves into the process of determining the value of the coefficient 'b' in a quadratic function when the axis of symmetry is known. Specifically, we will explore the function f(x) = (1/4)x² + bx + 10 and how to find the value of 'b' given that its axis of symmetry is x = 6. This exploration will not only enhance your understanding of quadratic functions but also equip you with the problem-solving skills necessary for tackling similar mathematical challenges.
Understanding Quadratic Functions and the Axis of Symmetry
To effectively address the problem at hand, it's essential to have a solid grasp of quadratic functions and the concept of the axis of symmetry. A quadratic function is a polynomial function of the second degree, generally expressed in the form f(x) = ax² + 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 upwards if 'a' is positive and downwards if 'a' is negative.
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 function reaches its minimum or maximum value. The x-coordinate of the vertex corresponds to the equation of the axis of symmetry. The formula for the axis of symmetry is given by x = -b / 2a, where 'a' and 'b' are the coefficients of the quadratic function. This formula is derived from completing the square or using calculus to find the critical point of the quadratic function. The axis of symmetry is a critical feature of the parabola because it allows us to easily identify the vertex and understand the symmetry of the graph. For instance, if we know the axis of symmetry and one point on the parabola, we can find another point equidistant from the axis of symmetry on the opposite side. This symmetry simplifies graphing and analyzing quadratic functions.
In the context of our problem, we are given the function f(x) = (1/4)x² + bx + 10 and the axis of symmetry x = 6. Our goal is to use the formula x = -b / 2a to find the value of 'b'. By understanding the relationship between the coefficients of the quadratic function and the axis of symmetry, we can solve for the unknown coefficient. This involves substituting the given values into the formula and solving the resulting equation. The process highlights the importance of understanding the properties of quadratic functions and how these properties can be used to solve mathematical problems.
Problem Setup: Identifying Given Information
Before diving into the solution, it's crucial to clearly identify the given information. We have the quadratic function f(x) = (1/4)x² + bx + 10, which is in the standard form f(x) = ax² + bx + c. By comparing the given function with the standard form, we can identify the coefficients: a = 1/4, b = b (which is what we need to find), and c = 10. We are also given that the axis of symmetry is x = 6. The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. This information is crucial because it directly relates to the coefficients of the quadratic function through the formula x = -b / 2a. The formula for the axis of symmetry provides a direct link between the coefficients 'a' and 'b' and the location of the axis of symmetry on the x-axis. Understanding this relationship is key to solving the problem.
By clearly identifying these pieces of information, we set the stage for a systematic approach to finding the value of 'b'. This step is essential in problem-solving as it helps to organize the given data and understand the connections between different elements of the problem. With the coefficients 'a' and the axis of symmetry known, we can now proceed to use the axis of symmetry formula to find the unknown coefficient 'b'. This involves substituting the known values into the formula and solving the resulting equation, which will be discussed in the next section.
Applying the Axis of Symmetry Formula
Now that we have identified the given information, we can apply the formula for the axis of symmetry to find the value of 'b'. The formula is x = -b / 2a, where x represents the axis of symmetry, and a and b are the coefficients of the quadratic function. We know that the axis of symmetry is x = 6 and the coefficient a = 1/4. Substituting these values into the formula, we get:
6 = -b / (2 * (1/4)).
This equation relates the unknown coefficient 'b' to the known values. The next step is to simplify and solve this equation for 'b'. First, we simplify the denominator: 2 * (1/4) = 1/2. So, the equation becomes:
6 = -b / (1/2).
To isolate 'b', we can multiply both sides of the equation by 1/2:
6 * (1/2) = -b.
This simplifies to:
3 = -b.
Finally, to solve for 'b', we multiply both sides by -1:
b = -3.
Therefore, the value of 'b' is -3. This result provides a specific value for the coefficient 'b' in the quadratic function f(x) = (1/4)x² + bx + 10. By applying the formula for the axis of symmetry and solving the resulting equation, we have successfully found the value of 'b' that satisfies the given conditions. This process demonstrates the practical application of the axis of symmetry formula in determining the parameters of a quadratic function. The next step is to verify the solution and ensure it aligns with the given information.
Verifying the Solution
After finding the value of b = -3, it's essential to verify the solution to ensure its accuracy. This involves substituting the calculated value of 'b' back into the original function and checking if the axis of symmetry matches the given value of x = 6. Our quadratic function now becomes:
f(x) = (1/4)x² - 3x + 10.
To find the axis of symmetry, we use the formula x = -b / 2a. Substituting the values a = 1/4 and b = -3 into the formula, we get:
x = -(-3) / (2 * (1/4))
Simplifying this expression:
x = 3 / (1/2)
x = 3 * 2
x = 6
As we can see, the calculated axis of symmetry x = 6 matches the given axis of symmetry. This confirms that our solution b = -3 is correct. Verification is a crucial step in problem-solving as it helps to avoid errors and ensures the accuracy of the final answer. By substituting the calculated value back into the original equation and checking against the given conditions, we can confidently conclude that our solution is valid. This step reinforces the importance of precision and attention to detail in mathematical problem-solving.
Conclusion
In conclusion, we have successfully determined the value of b in the quadratic function f(x) = (1/4)x² + bx + 10 given that its axis of symmetry is x = 6. By understanding the properties of quadratic functions and applying the formula for the axis of symmetry (x = -b / 2a), we were able to systematically solve for the unknown coefficient 'b'. The steps involved identifying the given information, substituting the values into the formula, solving the resulting equation, and verifying the solution. The final answer, b = -3, was confirmed by substituting it back into the function and verifying that the calculated axis of symmetry matched the given value.
This problem-solving process highlights the importance of a strong foundation in quadratic functions and their properties. Understanding the relationship between the coefficients and the axis of symmetry is crucial for solving various mathematical problems. Furthermore, the verification step underscores the importance of accuracy and attention to detail in mathematical calculations. By mastering these concepts and techniques, students can confidently tackle similar problems involving quadratic functions and their axes of symmetry. This comprehensive approach not only enhances problem-solving skills but also deepens the understanding of mathematical principles.
Quadratic function, axis of symmetry, vertex of parabola, coefficient 'b', solving for 'b', mathematical problem-solving, quadratic equations, parabola symmetry, function properties, verifying solutions.
