Finding The Value Of B Given The Axis Of Symmetry For A Quadratic Function
What is the value of b, given that the axis of symmetry for the graph of the function $f(x)=\frac{1}{4} x^2+b x+10$ is $x=6$?
In this article, we will explore how to determine the value of b in a quadratic function given its axis of symmetry. Specifically, we will focus on the function $f(x) = \frac{1}{4}x^2 + bx + 10$, where the axis of symmetry is $x = 6$. Understanding the relationship between the axis of symmetry and the coefficients of a quadratic function is crucial for solving various mathematical problems and gaining a deeper insight into quadratic equations and their graphical representations.
Understanding Quadratic Functions and the Axis of Symmetry
Let's start by defining what a quadratic function is and what the axis of symmetry represents. 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 reaches its minimum (if a > 0) or maximum (if a < 0) value. The equation of the axis of symmetry is given by $x = -\frac{b}{2a}$. This formula is derived from completing the square or using calculus to find the vertex of the parabola. The x-coordinate of the vertex is exactly the axis of symmetry.
The axis of symmetry plays a crucial role in understanding the behavior of a quadratic function. It helps us locate the vertex, which is a critical point for determining the minimum or maximum value of the function. Moreover, the symmetry property implies that for any two points on the parabola with the same y-coordinate, their x-coordinates are equidistant from the axis of symmetry. This symmetrical nature simplifies many problems related to quadratic functions, such as finding roots, determining intervals of increase and decrease, and sketching the graph. When dealing with real-world applications, such as projectile motion or optimization problems, understanding the axis of symmetry allows us to find the point of maximum height or minimum cost, respectively. The formula $x = -\frac{b}{2a}$ is a cornerstone in the study of quadratic functions, linking the algebraic coefficients to the geometric property of symmetry. By manipulating this formula, we can solve for unknown coefficients, given the axis of symmetry or vice versa. This connection between algebra and geometry is a powerful tool in mathematical problem-solving and provides a solid foundation for advanced topics in calculus and mathematical modeling. In practical scenarios, the axis of symmetry can represent the optimal point in various processes, from the trajectory of a ball thrown in the air to the ideal dimensions for maximizing the area of an enclosure with a fixed perimeter. Therefore, mastering the concept of the axis of symmetry is not only essential for academic success but also for real-world applications.
Applying the Axis of Symmetry Formula
In our given problem, we have the quadratic function $f(x) = \frac{1}{4}x^2 + bx + 10$. Comparing this to the general form $f(x) = ax^2 + bx + c$, we can identify that $a = \frac{1}{4}$, b is the unknown we need to find, and $c = 10$. We are also given that the axis of symmetry is $x = 6$. Using the formula for the axis of symmetry, $x = -\frac{b}{2a}$, we can set up an equation to solve for b. Substituting the given values, we get $6 = -\frac{b}{2(\frac{1}{4})}$. This equation relates the known axis of symmetry to the unknown coefficient b, allowing us to use algebraic manipulation to find the value of b.
To solve for b, we first simplify the denominator: $2(\frac1}{4}) = \frac{1}{2}$. The equation then becomes $6 = -\frac{b}{\frac{1}{2}}$. To eliminate the fraction in the denominator, we can multiply both sides of the equation by $\frac{1}{2}$ or recognize that dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can rewrite the equation as $6 = -2b$. Now, to isolate b, we divide both sides of the equation by -2{-2} = b$. This gives us the solution $b = -3$. Therefore, the value of b in the quadratic function $f(x) = \frac{1}{4}x^2 + bx + 10$ is -3, given that the axis of symmetry is $x = 6$. This result demonstrates how the formula for the axis of symmetry can be used to find unknown coefficients in a quadratic function. The process involves identifying the coefficients, substituting the known values into the formula, and solving the resulting equation for the unknown. This technique is a fundamental tool in algebra and is widely applicable in various mathematical contexts and real-world problems. Understanding this process is crucial for students and professionals alike, as it bridges the gap between algebraic expressions and their graphical representations. In addition, the ability to manipulate and solve equations involving fractions and algebraic expressions is a valuable skill that extends beyond quadratic functions and is essential for success in higher-level mathematics and related fields.
Solving for b
Now, let's solve the equation $6 = -\fracb}{2(\frac{1}{4})}$ step by step. First, we simplify the denominator4}) = \frac{1}{2}$. So the equation becomes $6 = -\frac{b}{\frac{1}{2}}$. Next, we multiply both sides of the equation by $\frac{1}{2}$ to get rid of the fraction in the denominator. Alternatively, we recognize that dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite the equation as $6 = -2b$. To isolate b, we divide both sides by -2{-2} = b$. This gives us $b = -3$.
The detailed step-by-step solution illustrates the importance of careful algebraic manipulation. Each step follows a logical progression, ensuring that the equation remains balanced and the solution is accurate. The ability to simplify fractions, multiply and divide both sides of an equation, and isolate the unknown variable are fundamental skills in algebra. These skills are not only essential for solving quadratic equations but also for tackling a wide range of mathematical problems. The process of solving for b in this context highlights the connection between algebraic equations and the properties of quadratic functions. By understanding how the axis of symmetry relates to the coefficients of the quadratic function, we can solve for unknown values and gain a deeper understanding of the function's behavior. Moreover, this process reinforces the importance of attention to detail and accuracy in mathematical calculations. A single error in any step can lead to an incorrect solution, emphasizing the need for careful and methodical problem-solving. In educational settings, this type of problem serves as an excellent exercise for students to practice their algebraic skills and apply their understanding of quadratic functions. It also provides a foundation for more advanced topics in mathematics, where the ability to manipulate equations and solve for unknowns is crucial.
Conclusion
Therefore, the value of b is -3. This problem demonstrates how the formula for the axis of symmetry can be used to find unknown coefficients in a quadratic function. By understanding the relationship between the coefficients and the axis of symmetry, we can solve various problems related to quadratic functions. In summary, to find the value of b given the axis of symmetry $x = 6$ for the function $f(x) = \frac{1}{4}x^2 + bx + 10$, we used the formula $x = -\frac{b}{2a}$, substituted the known values, and solved for b. This process not only provides the solution but also reinforces the importance of understanding the underlying principles of quadratic functions and their properties.
The ability to connect the algebraic representation of a quadratic function with its graphical characteristics is a key skill in mathematics. The axis of symmetry is a fundamental property that provides valuable insights into the behavior of the parabola. By mastering the relationship between the axis of symmetry and the coefficients of the quadratic function, students can solve a wide range of problems, including finding the vertex, determining the maximum or minimum value, and sketching the graph. Moreover, this understanding extends to real-world applications where quadratic functions are used to model various phenomena, such as projectile motion, optimization problems, and curve fitting. The process of solving for unknown coefficients, as demonstrated in this article, reinforces the importance of algebraic manipulation and problem-solving strategies. It also highlights the interconnectedness of different mathematical concepts and the need for a solid foundation in algebra to tackle more advanced topics. In conclusion, the value of b being -3 is not just a numerical answer but a result of a systematic application of mathematical principles and techniques, underscoring the power of mathematics in solving real-world problems and furthering our understanding of the world around us.