Evaluating F(x) = 2x^2 - 3x + 7 At X = 2: A Step-by-Step Guide

Have you ever wondered how to find the value of a function at a specific point? In mathematics, this is a fundamental skill, especially when dealing with quadratic functions. Today, we're going to dive deep into evaluating the function f(x) = 2x^2 - 3x + 7 when x = 2. This step-by-step guide will not only provide you with the solution but also help you understand the underlying concepts, ensuring you can tackle similar problems with confidence. Whether you're a student brushing up on your algebra or just curious about mathematical functions, this article will break down the process in a clear, easy-to-follow manner.

Understanding the Function

Before we jump into the calculation, let’s make sure we understand what the function f(x) = 2x^2 - 3x + 7 actually represents. This is a quadratic function, which is a type of polynomial function characterized by the highest power of x being 2. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In our case, a = 2, b = -3, and c = 7. The coefficients a, b, and c determine the shape and position of the parabola that represents the function when graphed. For instance, the coefficient a (in our case, 2) tells us that the parabola opens upwards because it's positive. If a were negative, the parabola would open downwards. Understanding the basic structure of the function is crucial because it helps us anticipate the behavior and properties of the function. Now that we have a grasp of the function's form, we can proceed to evaluate it at x = 2. Remember, evaluating a function at a specific point means substituting that value into the function and simplifying the expression to find the corresponding output value. This is a fundamental concept in algebra and calculus, and mastering it will be beneficial in various mathematical contexts.

Step-by-Step Evaluation

The core of our task is to evaluate f(x) = 2x^2 - 3x + 7 at x = 2. This means we're going to replace every instance of x in the function with the number 2. This process is called substitution, and it's a cornerstone of evaluating functions in mathematics. Let’s break it down step-by-step to ensure clarity and accuracy. First, we write down the function: f(x) = 2x^2 - 3x + 7. Next, we substitute x with 2: f(2) = 2(2)^2 - 3(2) + 7. Now we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We start with the exponent: 2 squared is 2 * 2 = 4. So, our equation becomes: f(2) = 2(4) - 3(2) + 7. Next, we perform the multiplication operations from left to right. 2 multiplied by 4 is 8, and -3 multiplied by 2 is -6. Now our equation looks like this: f(2) = 8 - 6 + 7. Finally, we perform the addition and subtraction operations from left to right. 8 minus 6 is 2, and 2 plus 7 is 9. Therefore, f(2) = 9. This tells us that when x is 2, the function f(x) outputs the value 9. Understanding this step-by-step process is vital because it ensures we don't make common arithmetic errors and that we follow the correct mathematical procedures. With practice, evaluating functions like this becomes second nature.

Detailed Calculation

Let's walk through the detailed calculation once more to solidify our understanding. We have the function f(x) = 2x^2 - 3x + 7, and we want to find f(2). The first step is substitution: replace every x with 2. This gives us f(2) = 2(2)^2 - 3(2) + 7. Now, we follow the order of operations (PEMDAS). Exponents come first, so we calculate 2 squared, which is 2 * 2 = 4. The equation now reads f(2) = 2(4) - 3(2) + 7. Next, we handle the multiplication. 2 multiplied by 4 is 8, and -3 multiplied by 2 is -6. So we have f(2) = 8 - 6 + 7. Now we move on to addition and subtraction, working from left to right. First, 8 minus 6 equals 2. This simplifies our equation to f(2) = 2 + 7. Finally, 2 plus 7 equals 9. Thus, we arrive at our answer: f(2) = 9. This detailed walkthrough emphasizes the importance of each step in the process. By carefully following the order of operations and double-checking our arithmetic, we can confidently arrive at the correct solution. It’s also worth noting that this method applies not only to quadratic functions but to any function you might encounter in mathematics. The key is to substitute the given value correctly and to follow the order of operations consistently.

Result and Interpretation

So, after our detailed calculations, we've determined that f(2) = 9 for the function f(x) = 2x^2 - 3x + 7. But what does this result actually mean? In mathematical terms, f(2) = 9 tells us that when the input value x is 2, the output value of the function f(x) is 9. Graphically, this corresponds to the point (2, 9) on the graph of the function. Imagine plotting the function on a coordinate plane; the point (2, 9) would lie on the curve representing the function. This is a crucial concept in understanding the relationship between inputs and outputs of a function. The input x is sometimes referred to as the independent variable, while the output f(x) is the dependent variable because its value depends on the value of x. The function itself is the rule that dictates how inputs are transformed into outputs. In this specific case, the rule is 2x^2 - 3x + 7. Understanding the graphical interpretation of function evaluation is also valuable. The point (2, 9) on the graph of f(x) is just one of infinitely many points that make up the curve. Each point represents a specific input-output pair, and collectively, they paint a picture of the function's behavior. By evaluating the function at various points, we can get a better sense of its overall shape and characteristics. This skill is particularly useful in fields like calculus, where understanding the behavior of functions is essential.

Conclusion

In conclusion, we have successfully evaluated the function f(x) = 2x^2 - 3x + 7 at x = 2 and found that f(2) = 9. We've walked through the process step-by-step, emphasizing the importance of substitution and following the order of operations (PEMDAS). This exercise not only provides a specific answer but also reinforces the fundamental concept of function evaluation, which is a cornerstone of algebra and calculus. By understanding how to substitute values into a function and simplify the resulting expression, you can confidently tackle a wide range of mathematical problems. Remember, the key is to break down the problem into smaller, manageable steps and to pay close attention to detail. Whether you're dealing with quadratic functions or more complex equations, the principles of function evaluation remain the same. With practice, this skill will become second nature, allowing you to explore and understand mathematical relationships more deeply. For further exploration of functions and their properties, you might find resources on websites like Khan Academy particularly helpful.