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

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Hey guys! Today, we're diving into a bit of math where we'll be evaluating a quadratic function. Specifically, we're going to look at the function f(x) = 2x² - 3x + 7. Imagine Liana is just starting to explore this function, and the first thing she wants to know is its value when x = 2. Sounds like fun, right? Let's get started and break down how to solve this step by step.

Understanding Quadratic Functions

Before we jump into the calculation, let's take a quick moment to understand what we're dealing with. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable x is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. In our case, a = 2, b = -3, and c = 7. These constants determine the shape and position of the parabola, which is the graph of a quadratic function. Quadratic functions pop up all over the place in math and real-world applications, from physics problems involving projectile motion to engineering designs for curved surfaces. They're a fundamental concept, and mastering them opens doors to solving a wide range of problems. Evaluating a quadratic function at a specific value of x simply means substituting that value for x in the function's expression and performing the arithmetic. This gives us the corresponding y-value, or f(x)-value, on the parabola. So, when we find f(2), we're essentially finding the y-coordinate of the point on the parabola where the x-coordinate is 2. This point gives us valuable information about the function's behavior at that specific x-value. It can tell us if the function is increasing or decreasing, and it helps us understand the overall trend of the parabola. The process of evaluating functions is a core skill in mathematics, and it's something you'll use frequently as you delve deeper into algebra and calculus. It lays the foundation for more complex concepts, such as finding roots, determining the vertex, and graphing functions. So, let's get back to our specific problem and see how Liana can find the value of f(x) when x = 2. We'll break it down step by step to make it super clear and easy to follow, even if you're just starting to learn about quadratic functions. Remember, practice makes perfect, so the more you work with these functions, the more comfortable you'll become. Okay, let's do this!

Step-by-Step Evaluation of f(2)

Okay, let's get down to business and actually calculate the value of f(2) for our function f(x) = 2x² - 3x + 7. The key here is substitution: we're going to replace every instance of x in the function's expression with the number 2. This might sound simple, but it's crucial to do it carefully to avoid any errors. Here’s how we do it step by step:

  1. Substitute x with 2: First, we rewrite the function, replacing x with 2 in parentheses. This helps us keep track of where we've made the substitution and makes the next steps clearer. So, we get:

    f(2) = 2(2)² - 3(2) + 7

    Notice how we've put 2 in parentheses everywhere x used to be. This is especially important when dealing with negative numbers or exponents, as it ensures we apply the operations in the correct order.

  2. Evaluate the exponent: Next up, we need to handle the exponent. Remember the order of operations (PEMDAS/BODMAS)? Exponents come before multiplication and addition, so we tackle that first. 2 squared, or 2², is simply 2 multiplied by itself, which equals 4. So, we replace 2² with 4 in our expression:

    f(2) = 2(4) - 3(2) + 7

    Now things are looking a little simpler, right? We've gotten rid of the exponent and are left with multiplication, subtraction, and addition.

  3. Perform the multiplications: Now, let's move on to the multiplications. We have two multiplications to take care of: 2 times 4 and -3 times 2. Let's do them one at a time:

    2(4) = 8 -3(2) = -6

    So, we replace those products in our expression, and we get:

    f(2) = 8 - 6 + 7

    We're getting closer! All that's left now are some simple addition and subtraction operations.

  4. Perform the addition and subtraction: Finally, we perform the addition and subtraction from left to right. This is crucial to ensure we get the correct answer. So, first, we do 8 minus 6:

    8 - 6 = 2

    Then, we add 7 to the result:

    2 + 7 = 9

    And there you have it! We've successfully evaluated f(2).

The Final Result: f(2) = 9

So, after all those steps, we've arrived at our final answer: f(2) = 9. This means that when x is equal to 2, the value of the function f(x) = 2x² - 3x + 7 is 9. In terms of the graph of the function, this tells us that the point (2, 9) lies on the parabola. Understanding how to evaluate functions like this is a fundamental skill in algebra and beyond. It allows us to understand how the function behaves for different input values and forms the basis for more complex concepts like graphing and solving equations. If Liana (or you!) were to plot this point on a graph, it would be a single dot on the curve that represents the function. But remember, this is just one point on the parabola. To get a full picture of the function, you'd need to evaluate it for many different values of x and plot those points. Then you could connect the dots to see the shape of the curve. Evaluating functions is also important in real-world applications. For example, if this function represented the height of a ball thrown in the air, then f(2) = 9 would tell us that the ball is 9 units high after 2 seconds. These types of applications make understanding functions even more crucial. The process of substitution, simplifying exponents, performing multiplication, and finally, handling addition and subtraction is the core of evaluating any polynomial function, not just quadratic ones. It's a systematic approach that, once mastered, can be applied to more complex expressions as well. So, if you ever encounter another function and need to find its value at a specific point, remember these steps. Break it down, be methodical, and you'll get there! Don't be intimidated by the symbols or the equations. Just take it one step at a time, and you'll see that it's all quite manageable. And remember, the more you practice, the easier it becomes. So keep exploring, keep evaluating, and keep having fun with math!

Key Takeaways

Before we wrap things up, let's just quickly recap the key takeaways from this exercise. This will help solidify our understanding and make sure we've grasped the core concepts. First and foremost, we learned how to evaluate a quadratic function at a specific point. This involves substituting the given value for x into the function's expression and then simplifying using the order of operations. We saw how important it is to follow the correct order (PEMDAS/BODMAS) to ensure we arrive at the accurate answer. Exponents need to be dealt with before multiplication, and multiplication before addition and subtraction. We also highlighted the importance of careful substitution, especially when dealing with negative numbers or exponents. Using parentheses can be a lifesaver in preventing errors. We emphasized that evaluating functions is a fundamental skill in mathematics that forms the basis for more advanced concepts. It's not just about plugging in numbers; it's about understanding how a function behaves for different inputs. We also touched upon the connection between evaluating functions and their graphical representation. The value f(2) = 9 corresponds to a specific point (2, 9) on the parabola of the quadratic function. Evaluating functions helps us understand the shape and behavior of the graph. Furthermore, we briefly discussed the real-world applications of function evaluation. Many physical phenomena can be modeled by functions, and evaluating them allows us to make predictions and solve problems. Finally, we reiterated that the steps we used to evaluate this quadratic function can be generalized to other polynomial functions as well. The systematic approach of substitution and simplification is a powerful tool in mathematics. So, keep these key takeaways in mind as you continue your mathematical journey. Remember, practice is key to mastery. The more you work with functions and evaluate them, the more confident and proficient you'll become. Keep exploring, keep asking questions, and keep having fun with math!

I hope this breakdown was helpful and that Liana's evaluation of f(2) is now crystal clear! Remember, math is all about practice and understanding the steps. Keep at it, and you'll be solving these problems like a pro in no time!