Evaluating F(x) = -5x^2 + 2x + 9: Find F(1) And F(2)

Hey guys! Today, we're diving into the world of functions, specifically looking at how to evaluate a quadratic function at different points. We've got the function f(x) = -5x^2 + 2x + 9, and our mission is to find the values of f(1) and f(2). Once we've done that, we'll compare these values and see which statement about them holds true. Let's jump right in!

Finding f(1):

Okay, so the first step is to figure out what f(1) is. What does f(1) mean? It means we need to substitute x = 1 into our function. So, everywhere we see an x in the function, we're going to replace it with a 1. Let's do it!

f(1) = -5(1)^2 + 2(1) + 9

Now we need to follow the order of operations (PEMDAS/BODMAS), which means we tackle exponents first.

f(1) = -5(1) + 2(1) + 9

Next up is multiplication:

f(1) = -5 + 2 + 9

And finally, addition and subtraction from left to right:

f(1) = -3 + 9

f(1) = 6

Awesome! We've found that f(1) = 6. That wasn't too bad, right? Now, let's move on to finding f(2). Remember, understanding function notation is key here; it's all about substituting the given value into the function and simplifying. This skill is fundamental in algebra and calculus, so mastering it now will definitely pay off later. Think of functions as machines: you put in a number (x), and the machine spits out another number (f(x)) based on the function's rule. In this case, our machine squares the input, multiplies it by -5, adds twice the input, and then adds 9. It's like a mathematical recipe! The process of evaluating functions is not just a mechanical exercise; it provides a powerful way to model real-world phenomena. For instance, the height of a ball thrown in the air can be modeled by a quadratic function, similar to the one we're working with. By evaluating the function at different times (x values), we can determine the ball's height at those times (f(x) values). This ability to connect mathematical functions to real-world situations is what makes them so valuable in science and engineering.

Finding f(2):

Alright, now let's tackle f(2). Just like before, we're going to substitute x = 2 into our function f(x) = -5x^2 + 2x + 9. So, let's replace all the x's with 2's.

f(2) = -5(2)^2 + 2(2) + 9

Again, we start with exponents:

f(2) = -5(4) + 2(2) + 9

Then we multiply:

f(2) = -20 + 4 + 9

And finally, we add and subtract from left to right:

f(2) = -16 + 9

f(2) = -7

Boom! We've found that f(2) = -7. See? You're getting the hang of this! Now we have both f(1) and f(2). The beauty of function evaluation lies in its consistency and predictability. Once you understand the rule of the function, you can plug in any value for x and get a corresponding output. This consistency is what allows us to create graphs of functions, which visually represent the relationship between x and f(x). Imagine plotting the points (1, 6) and (2, -7) on a graph. These are just two points on the parabola defined by our quadratic function. By plotting more points, we can get a clearer picture of the function's behavior. Moreover, understanding how to evaluate functions is crucial for more advanced mathematical concepts like calculus. In calculus, we often need to find the limit of a function as x approaches a certain value, which involves evaluating the function at values very close to that point. So, mastering function evaluation now will set you up for success in future math courses.

Comparing f(1) and f(2):

Okay, we've done the heavy lifting. We know that f(1) = 6 and f(2) = -7. Now it's time to compare these two values. Which one is bigger? Think about it for a second...

Since 6 is a positive number and -7 is a negative number, 6 is definitely larger than -7. So, f(1) is greater than f(2).

Now, let's look back at the statements provided and see which one is true:

• A. The value of f(1) cannot be compared to the value of f(2).
• B. The value of f(2) is larger than the value of f(1).

We know that statement A is false because we just compared the two values. We also know that statement B is false because f(1) is larger than f(2). In this specific problem, the provided choices did not include the correct statement but based on our calculations, we can conclude f(1) is larger than f(2).

Comparing values is a fundamental skill in mathematics. Understanding inequalities is crucial for solving many types of problems, from simple comparisons like this one to more complex optimization problems. Think about it: we often need to compare different options and choose the best one, whether it's the cheapest, the fastest, or the most efficient. Mathematical inequalities provide the tools to make these comparisons in a precise and logical way. Furthermore, the ability to compare function values is essential for understanding the behavior of functions. Is the function increasing or decreasing? Where does it reach its maximum or minimum value? These are the types of questions we can answer by comparing function values at different points. In calculus, we use derivatives to analyze the rate of change of a function, which is essentially a sophisticated way of comparing function values over very small intervals. So, the simple act of comparing f(1) and f(2) lays the groundwork for more advanced mathematical concepts.

Conclusion:

So, there you have it! We successfully evaluated the function f(x) = -5x^2 + 2x + 9 at x = 1 and x = 2, found that f(1) = 6 and f(2) = -7, and determined that f(1) is larger than f(2). Great job, guys! Remember, practice makes perfect, so keep plugging away at these types of problems, and you'll be a function-evaluating pro in no time!

This exercise highlights the importance of careful calculation and attention to detail. It's easy to make a small mistake, like forgetting a negative sign or misapplying the order of operations, which can lead to a wrong answer. Always double-check your work, especially when dealing with negative numbers and exponents. Function evaluation is a cornerstone of algebra and calculus. By mastering this skill, you're building a solid foundation for future mathematical endeavors. So, keep practicing, keep exploring, and keep having fun with math! Remember that each problem you solve is a step forward on your mathematical journey. The more you practice, the more confident and skilled you'll become. And who knows, maybe one day you'll be using these skills to solve real-world problems in science, engineering, or even finance. The possibilities are endless!