Hey guys! Let's dive into a fun math problem that involves understanding the axis of symmetry of a quadratic function. This concept is super important in algebra and helps us visualize and analyze parabolas, the U-shaped curves that quadratic functions create. So, let's break it down step by step.
What is the Axis of Symmetry?
First off, what exactly is the axis of symmetry? Imagine you have a parabola, like the one formed by the function f(x) = (1/4)x^2 + bx + 10 in our problem. The axis of symmetry is an imaginary vertical line that cuts the parabola perfectly in half. It's like a mirror; whatever is on one side of the line is mirrored exactly on the other side. This line is crucial because it tells us where the vertex (the highest or lowest point) of the parabola is located. And knowing the vertex is key to understanding the entire shape and behavior of the quadratic function. The axis of symmetry always passes through the vertex of the parabola. For a quadratic function in the standard form of f(x) = ax^2 + bx + c, the equation for the axis of symmetry is given by x = -b / 2a. This formula is your best friend when dealing with these types of problems. It directly links the coefficients of the quadratic equation to the location of the axis of symmetry. Understanding this relationship is super helpful for solving problems efficiently and accurately. So, remember this formula – it's a game-changer! Now, let's think about why the axis of symmetry is so significant. It not only helps us find the vertex but also provides valuable information about the parabola's symmetry. If you know one point on the parabola, you immediately know its mirror image on the other side of the axis. This symmetry simplifies graphing and analyzing quadratic functions considerably. For example, if you find that a point (x1, y1) lies on the parabola, then the point (2h - x1, y1) will also lie on the parabola, where x = h is the equation of the axis of symmetry. This property can save you a lot of time when you're plotting points or trying to understand the function's behavior. In the context of real-world applications, the axis of symmetry can help us find maximum or minimum values. For instance, if you're modeling the trajectory of a ball thrown in the air, the vertex represents the highest point the ball reaches, and the axis of symmetry tells you the time at which the ball reaches that height. Similarly, in business, quadratic functions can model profit curves, where the vertex represents the maximum profit, and the axis of symmetry indicates the input level that maximizes profit. So, understanding the axis of symmetry is not just a theoretical concept; it has practical applications in various fields.
Our Function: f(x) = (1/4)x^2 + bx + 10
Now, let's look closely at our specific function: f(x) = (1/4)x^2 + bx + 10. In this quadratic equation, we have 'a', 'b', and 'c' coefficients. Remember, the general form of a quadratic function is f(x) = ax^2 + bx + c. So, in our case: 'a' is 1/4, 'b' is the value we need to find, and 'c' is 10. The coefficient 'a' (1/4 in our case) determines whether the parabola opens upwards or downwards. Since 'a' is positive, our parabola opens upwards, meaning it has a minimum point (a vertex at the bottom). If 'a' were negative, the parabola would open downwards, having a maximum point (a vertex at the top). The magnitude of 'a' also affects how wide or narrow the parabola is. A smaller 'a' value (closer to zero) makes the parabola wider, while a larger 'a' value makes it narrower. The coefficient 'b' plays a crucial role in determining the position of the axis of symmetry and, consequently, the vertex of the parabola. As we discussed earlier, the axis of symmetry is given by the formula x = -b / 2a. Changing the value of 'b' shifts the parabola horizontally. If 'b' is positive, the parabola shifts to the left, and if 'b' is negative, it shifts to the right. The constant term 'c' (10 in our function) represents the y-intercept of the parabola. This is the point where the parabola crosses the y-axis. In other words, it's the value of f(x) when x = 0. So, for our function, the parabola intersects the y-axis at the point (0, 10). Understanding these coefficients and their effects on the parabola is fundamental to solving quadratic function problems. By analyzing 'a', 'b', and 'c', we can sketch a rough graph of the parabola, identify its key features, and solve related problems efficiently. In our case, we know 'a' and 'c', and we are given information about the axis of symmetry, which will help us determine the value of 'b'. So, let's move on to the next step and use the given information to find 'b'. Remember, math is like a puzzle, and each piece of information we have helps us fit the puzzle together and find the solution!
Using the Given Axis of Symmetry: x = 6
Okay, here's the key piece of information: the axis of symmetry is x = 6. This tells us that the vertical line that cuts our parabola in half is located at x = 6. Remember the formula we talked about? The axis of symmetry for a quadratic function f(x) = ax^2 + bx + c is given by x = -b / 2a. We know that 'a' is 1/4 and the axis of symmetry is 6. We can plug these values into the formula and solve for 'b'. This is where the magic happens! By substituting the known values into the formula, we create an equation that we can solve for our unknown, 'b'. This is a common technique in algebra: using given information to form an equation and then solving for the variable we're interested in. The formula x = -b / 2a is derived from the process of completing the square or by finding the vertex of the parabola using calculus. It's a powerful tool that connects the coefficients of the quadratic function to a key geometric feature, the axis of symmetry. Now, let's think about what the axis of symmetry tells us about the vertex of the parabola. Since the vertex lies on the axis of symmetry, we know that the x-coordinate of the vertex is 6. This is a valuable piece of information because the vertex represents the minimum (or maximum) value of the quadratic function. In our case, since the parabola opens upwards (because 'a' is positive), the vertex represents the minimum value of the function. To find the y-coordinate of the vertex, we would substitute x = 6 into our function f(x) = (1/4)x^2 + bx + 10. However, we don't need to find the y-coordinate to solve for 'b'. The x-coordinate and the formula for the axis of symmetry are enough to get us there. So, let's set up the equation using the formula x = -b / 2a and the given information. We have 6 = -b / (2 * (1/4)). Now, it's just a matter of solving this equation for 'b'. We'll multiply both sides by 2 * (1/4) and then solve for 'b'. This is a straightforward algebraic manipulation, but it's crucial to get the steps right to arrive at the correct answer. So, let's be careful and methodical as we proceed with the calculations.
Solving for b
Let's solve for 'b' step by step. We have the equation 6 = -b / (2 * (1/4)). First, simplify the denominator: 2 * (1/4) = 1/2. So, our equation becomes 6 = -b / (1/2). To get rid of the fraction in the denominator, we can multiply both sides of the equation by 1/2: 6 * (1/2) = -b / (1/2) * (1/2). This simplifies to 3 = -b / 2. Now, to isolate 'b', we can multiply both sides by -2: 3 * -2 = (-b / (1/2)) * -2. This gives us -6 = b. So, we've found that b = -3. Isn't it satisfying when the pieces of the puzzle come together? We started with a quadratic function and the axis of symmetry, and through careful application of the formula and algebraic manipulation, we determined the value of 'b'. This is a great example of how math can be used to solve real problems and how understanding the underlying concepts can make the process much easier. Now, let's double-check our answer to make sure it makes sense in the context of the original problem. We found that b = -3, so our function is f(x) = (1/4)x^2 - 3x + 10. The axis of symmetry is given by x = -b / 2a, which in our case is x = -(-3) / (2 * (1/4)) = 3 / (1/2) = 6. This matches the given information, so we can be confident that our answer is correct. Moreover, we can think about the shape of the parabola with b = -3. Since 'a' is positive (1/4), the parabola opens upwards, and since 'b' is negative, the axis of symmetry is shifted to the right of the y-axis, which is consistent with the axis of symmetry being x = 6. This kind of reasoning helps us develop a deeper understanding of quadratic functions and their graphs. So, remember, solving math problems is not just about finding the right answer; it's also about understanding the process and the concepts involved. Each problem is an opportunity to learn something new and improve your problem-solving skills.
Final Answer
Therefore, the value of b is -3. We did it! By understanding the concept of the axis of symmetry and applying the correct formula, we were able to solve for the unknown coefficient 'b' in our quadratic function. This problem highlights the importance of knowing key formulas and how to manipulate them to solve for different variables. Remember, guys, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time!