Solving For 'b': Axis Of Symmetry Problem

Hey math enthusiasts! Let's dive into a classic algebra problem that combines quadratic functions and the concept of the axis of symmetry. We're given a quadratic function, and our mission is to find the value of a specific coefficient. Sounds fun, right? Buckle up, because we're about to break down this problem in a way that's easy to understand. We'll explore the axis of symmetry, its relationship with the quadratic formula, and ultimately, how to solve for our unknown variable, 'b'.

Understanding the Axis of Symmetry

So, what exactly is the axis of symmetry? Imagine a perfectly symmetrical shape, like a butterfly. The axis of symmetry is the imaginary line that divides the shape into two identical halves. In the context of a quadratic function (which, when graphed, forms a U-shaped curve called a parabola), the axis of symmetry is a vertical line that passes through the vertex (the lowest or highest point) of the parabola. This line essentially cuts the parabola in half, making the two sides mirror images of each other. The equation of the axis of symmetry is always in the form of x = some constant. That constant represents the x-coordinate of the vertex.

For a quadratic function written in the standard form f(x) = ax² + bx + c, the axis of symmetry can be calculated using the formula x = -b / 2a. This formula is derived directly from completing the square or by understanding the properties of the parabola's vertex. The 'a' and 'b' in the formula come from the coefficients in the quadratic equation. Understanding this is key because it gives us a direct link between the equation of the parabola and the location of its axis of symmetry. The axis of symmetry is a fundamental concept when working with quadratic equations, as it helps identify the vertex, understand the roots or zeros of the equation, and describe the parabola's behavior.

Now, let's connect this to our specific problem. We are told that the axis of symmetry for the graph of the function f(x) = (1/4)x² + bx + 10 is x = 6. This means that the vertical line x = 6 divides the parabola into two symmetrical halves, and it also passes through the vertex. Our goal is to use this information, combined with the formula for the axis of symmetry, to solve for 'b'. The value of 'b' is a crucial coefficient in the quadratic equation, which influences the position and shape of the parabola. Determining the correct value of 'b' allows us to accurately graph and analyze the function.

Applying the Axis of Symmetry Formula to Solve for 'b'

Now, let's get down to the actual solving part. We know the axis of symmetry (x = 6) and we know the general formula for the axis of symmetry (x = -b / 2a). Our function is f(x) = (1/4)x² + bx + 10. Comparing this to the standard form f(x) = ax² + bx + c, we can see that:

• a = 1/4
• The 'b' in our function is the 'b' we need to solve for.
• c = 10 (but we don't need 'c' for this particular problem)

We can plug these values into the axis of symmetry formula: x = -b / 2a.

We know that x = 6, and a = 1/4. So the equation becomes: 6 = -b / (2 * (1/4)).

Let's simplify that a bit. 2 * (1/4) = 1/2. So, the equation is now: 6 = -b / (1/2).

To solve for 'b', we need to isolate it. 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 means that when b = -3, the axis of symmetry of the parabola represented by the equation f(x) = (1/4)x² - 3x + 10 is indeed x = 6. This process highlights how the axis of symmetry provides a direct method for finding unknown coefficients within quadratic equations. Remember, the axis of symmetry is always a powerful tool. It gives us a peek into the inner workings of the equation and its corresponding graph. Using this method, we can quickly discover critical values, like the vertex of the parabola, its roots, or, in this case, a crucial coefficient like 'b'. By connecting the given information to the appropriate formula, and a little bit of algebraic manipulation, we successfully cracked the code and found our solution.

Confirming the Solution and Final Thoughts

We found that b = -3. To ensure our answer is correct, let's plug it back into the original equation and recalculate the axis of symmetry to verify it matches x = 6. If b = -3, our equation becomes f(x) = (1/4)x² - 3x + 10. Using the formula x = -b / 2a, with a = 1/4 and b = -3, we get: x = -(-3) / (2 * (1/4)). This simplifies to x = 3 / (1/2), which equals x = 6.

This confirms that our answer is correct! The axis of symmetry is indeed x = 6 when b = -3. We can be confident that our calculations are accurate and that we have successfully solved the problem. Understanding and using the axis of symmetry is a fundamental skill in algebra, which enables us to tackle various problems related to quadratic equations and their graphs. This includes determining the vertex, understanding the behavior of the parabola, and finding solutions or roots. Through the process of understanding the concept, the formula, and applying it, this problem becomes very manageable.

Mastering these concepts isn't just about getting the right answer; it's about building a solid foundation in mathematics. So, next time you encounter a problem involving a quadratic function and its axis of symmetry, remember the steps we've covered today. Remember to analyze the problem, use the proper formula, and systematically solve for the unknown variables. Keep practicing, keep exploring, and keep the curiosity alive! Mathematics is not just a collection of formulas, but a way of thinking, a way of solving problems, and a way of understanding the world around us. Keep on studying, and happy calculating!