Hey guys! Today, we're diving into the world of functions, specifically evaluating a function at a given point. We've got a fun one here: What is f(-3) for the function f(a) = -2a² - 5a + 4? Don't worry; it's not as scary as it looks. We'll break it down step-by-step, so you'll be a function-evaluating pro in no time!
Understanding Function Evaluation
First, let's make sure we're all on the same page about what it means to evaluate a function. A function, in simple terms, is like a machine. You put something in (the input), and the machine does something to it and spits out something else (the output). In our case, our function is f(a) = -2a² - 5a + 4. The 'a' is the input variable, and the function tells us exactly what to do with that input. Function evaluation means we're plugging in a specific value for the input variable (in this case, 'a') and calculating the output. Think of it like replacing 'a' with the number -3 everywhere it appears in the function's formula. This might sound a bit abstract, but it's a fundamental concept in mathematics, especially when you start exploring more advanced topics like calculus and real analysis. The ability to confidently evaluate functions is a cornerstone of mathematical proficiency and will serve you well in various mathematical contexts.
When we talk about the function f(a) = -2a² - 5a + 4, it's crucial to understand the different parts and what they represent. The 'f' is the name of the function. We could have called it 'g', 'h', or anything else, but 'f' is a pretty standard choice. The '(a)' tells us that the input variable is 'a'. This means that whatever we put inside the parentheses, that's what we're going to substitute for 'a' in the rest of the expression. The expression '-2a² - 5a + 4' is the actual rule that the function follows. It tells us what to do with the input 'a'. We first square it, then multiply by -2. Then, we multiply 'a' by -5. Finally, we add 4 to the whole thing. This order of operations is critical; follow it carefully to get the correct answer! So, when we want to find f(-3), we're essentially asking the function machine: "Hey, what do you do when I give you -3 as an input?"
Step-by-Step Solution for f(-3)
Okay, let's get our hands dirty and actually solve this thing! We want to find f(-3), which means we're going to substitute -3 for 'a' in the function f(a) = -2a² - 5a + 4. Here's how it looks:
f(-3) = -2(-3)² - 5(-3) + 4
Now, we need to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). The key here is to be meticulous and take it one step at a time. Rushing through can lead to silly mistakes, especially with negative signs. Remember, a small error early on can throw off the entire calculation, so accuracy is paramount.
First up: Exponents! We have (-3)². Remember, this means -3 multiplied by itself: (-3) * (-3) = 9. So, we can rewrite our expression as:
f(-3) = -2(9) - 5(-3) + 4
Next, we tackle the multiplications. We have two multiplications to perform: -2(9) and -5(-3). Let's do them one by one. -2 multiplied by 9 is -18. -5 multiplied by -3 is +15 (remember, a negative times a negative is a positive!). Our expression now looks like this:
f(-3) = -18 + 15 + 4
Finally, we perform the additions from left to right. -18 + 15 equals -3. Then, we add 4: -3 + 4 = 1. So, after all that, we have:
f(-3) = 1
Common Mistakes to Avoid
Now that we've walked through the solution, let's talk about some common pitfalls that students often encounter when evaluating functions. Being aware of these mistakes can help you avoid them and ensure you get the correct answer. Spotting these errors before they happen is a skill that comes with practice and a solid understanding of the underlying concepts.
One of the most frequent errors is messing up the order of operations. Remember PEMDAS/BODMAS! Exponents come before multiplication, which comes before addition and subtraction. If you don't follow this order, you're almost guaranteed to get the wrong answer. For example, in our problem, if you were to multiply -2 by -3 before squaring -3, you'd end up with a completely different result. Another common mistake involves dealing with negative signs. It's crucial to be careful when squaring negative numbers and when multiplying negative numbers together. A negative times a negative is a positive, and a negative times a positive is a negative. These rules are fundamental, and a slip-up here can change the sign of your answer, leading to an incorrect solution. For instance, forgetting that (-3)² = 9 and instead treating it as -9 would throw off the entire calculation. Finally, be mindful of careless arithmetic errors. Even if you understand the process perfectly, a simple addition or subtraction mistake can lead to the wrong answer. It's always a good idea to double-check your work, especially when dealing with multiple steps. Write out each step clearly and systematically to minimize the chance of making a mistake.
Practice Makes Perfect
The best way to master function evaluation is, you guessed it, practice! Try plugging in different values for 'a' in our function f(a) = -2a² - 5a + 4. What's f(0)? What's f(2)? What about f(-1)? The more you practice, the more comfortable you'll become with the process. Consistent practice builds confidence and reinforces the concepts in your mind. It also helps you develop a sense of the patterns and relationships within the function. You can start with simple integers and then gradually move on to fractions, decimals, and even variables to challenge yourself further.
You can also try evaluating different types of functions. We worked with a quadratic function (one with an a² term), but there are linear functions (like f(a) = 3a + 2), cubic functions (with an a³ term), and many others. Each type of function has its own unique characteristics, but the basic principle of evaluation remains the same: substitute the given value for the variable and simplify. Consider exploring online resources, textbooks, or worksheets that offer a variety of function evaluation problems. Working through these problems will not only solidify your understanding but also expose you to different problem-solving strategies and techniques. Remember, the goal is not just to get the right answer but to develop a deep conceptual understanding of how functions work. This understanding will serve you well as you progress in your mathematical journey.
Conclusion
So, there you have it! We've successfully evaluated f(-3) for the function f(a) = -2a² - 5a + 4. The answer, as we found, is 1. We've walked through the process step-by-step, discussed common mistakes to avoid, and emphasized the importance of practice. Evaluating functions is a crucial skill in mathematics, and with a little bit of effort, you can master it. Keep practicing, and you'll be a function whiz in no time!
