Finding H(-8) Given H(t) = -2(t+5)^2 + 4 A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate a function. Specifically, we're given the function h(t) = -2(t+5)^2 + 4, and our mission, should we choose to accept it (and we totally do!), is to find the value of h(-8). In simpler terms, this means we need to plug in -8 for t in the equation and then carefully calculate the result. Now, before we jump into the nitty-gritty calculations, let's take a moment to understand what this function actually represents. The function h(t) is a quadratic function, which means when we graph it, we'll get a parabola – a U-shaped curve. The t represents the input, and h(t) represents the output, or the y-value on our graph. Understanding the function's form can sometimes give us clues about its behavior, but for this particular problem, we're mainly focused on the arithmetic. So, let’s get started!

Step-by-Step Solution: Calculating h(-8)

Okay, so the first thing we need to do is substitute -8 for every t we see in the function. This gives us h(-8) = -2((-8)+5)^2 + 4. Notice how we've carefully replaced t with -8, making sure to keep the parentheses intact. This is super important because the order of operations (PEMDAS/BODMAS) dictates that we deal with what's inside the parentheses first. So, let's tackle that next. Inside the parentheses, we have -8 + 5. What does that equal? It's a simple addition, and the answer is -3. So now our equation looks like this: h(-8) = -2(-3)^2 + 4. We're making progress! The next operation in the order of operations is the exponent. We need to square -3. Remember, squaring a number means multiplying it by itself. So, -3 squared (-3 * -3) is positive 9. Keep in mind that a negative number squared becomes positive because a negative times a negative is a positive. Now our equation transforms to h(-8) = -2(9) + 4. We're getting closer to the finish line!

Now, according to the order of operations, we need to perform multiplication before addition. So, we multiply -2 by 9. This gives us -18. Our equation now reads h(-8) = -18 + 4. Almost there! Finally, we perform the addition: -18 + 4. If we're starting at -18 on the number line and move 4 places to the right, we end up at -14. Thus, h(-8) = -14. And that, my friends, is our final answer! We've successfully evaluated the function h(t) at t = -8. This whole process demonstrates the importance of following the order of operations precisely. If we'd accidentally multiplied before squaring, or added before multiplying, we would have ended up with a completely different, and incorrect, answer. Math is all about precision, guys!

Understanding the Implications of h(-8) = -14

So, we've figured out that h(-8) = -14. But what does this actually mean in the grand scheme of things? Well, remember how we talked about h(t) representing a parabola? This result gives us a specific point on that parabola. The input, -8, is our x-coordinate, and the output, -14, is our y-coordinate. So, the point (-8, -14) lies on the graph of the function h(t). Visualizing this can be really helpful. Imagine a U-shaped curve, and somewhere on that curve, there's a point where the x-value is -8 and the y-value is -14. This point is just one tiny piece of the overall picture of the function, but it's an important piece nonetheless. We can find many other points on the parabola by plugging in different values for t. Each point gives us a snapshot of the function's behavior at a specific input. When we have enough points, we can start to see the overall shape and trend of the parabola. The vertex, which is the highest or lowest point on the parabola, is a particularly important feature. In our case, since the coefficient of the squared term is negative (-2), the parabola opens downward, meaning the vertex is the highest point. Finding the vertex and a few other key points allows us to sketch the graph of the function fairly accurately.

Furthermore, knowing h(-8) = -14 can be useful in various applications. For instance, if this function represented the height of a projectile (like a ball thrown in the air) at different times, then h(-8) = -14 would tell us that at 8 seconds before the projectile was launched (since time can be negative in mathematical models), its height was -14 units. Now, in the real world, a negative height might not make physical sense (unless we're talking about something below ground level), but in the mathematical model, it's a perfectly valid point on the graph. The key takeaway here is that function evaluation is a fundamental skill in mathematics and has applications in many different fields. Whether we're analyzing physical phenomena, modeling economic trends, or designing engineering structures, functions play a crucial role. Mastering the ability to plug in values and calculate outputs is essential for anyone working with mathematical models.

Common Mistakes to Avoid When Evaluating Functions

Okay, guys, let's talk about some common pitfalls that people often stumble into when evaluating functions. Knowing these mistakes can help us steer clear of them and ensure we get the correct answer every time. One of the biggest culprits, as we mentioned earlier, is messing up the order of operations. It's super tempting to just go from left to right, doing whatever operation comes next, but that's a recipe for disaster. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Stick to this order like glue, and you'll be in good shape. Another frequent mistake is related to negative signs. They can be tricky little devils! For example, when we square a negative number, like we did with -3 earlier, it becomes positive. Forgetting that a negative times a negative is a positive is a common blunder. Similarly, be extra careful when distributing a negative sign across parentheses. Make sure you apply the negative to every term inside the parentheses. A third area where errors often creep in is with arithmetic. Simple addition, subtraction, multiplication, and division can sometimes trip us up, especially when dealing with negative numbers or fractions. Double-check your calculations, and don't be afraid to use a calculator if you need to. It's better to be safe than sorry!

Finally, sometimes people make mistakes simply by not writing things down clearly. Math can be a bit like a detective story – you need to show your work so you can follow your own reasoning and spot any errors. If you try to do too much in your head, it's easy to lose track of a negative sign or make a small arithmetic mistake. Writing each step clearly and systematically makes it much easier to catch those errors. So, in a nutshell, avoid order of operations errors, watch out for those sneaky negative signs, double-check your arithmetic, and always show your work. By being mindful of these common mistakes, you'll dramatically improve your accuracy when evaluating functions.

Practice Problems: Test Your Understanding

Alright, now that we've gone through the solution and discussed some common pitfalls, it's time to put your newfound skills to the test! Practice makes perfect, guys, and the more you work with function evaluation, the more comfortable and confident you'll become. So, here are a few practice problems for you to try. Grab a piece of paper and a pencil, and let's get started!

1. Given f(x) = 3x^2 - 2x + 1, find f(2).
2. If g(t) = -t^3 + 4t - 5, evaluate g(-1).
3. Let h(x) = 5(x - 3)^2 + 2. Find h(3).
4. Suppose p(t) = -2|t + 1| + 7. Evaluate p(-4) (remember that | | denotes absolute value).
5. Consider the function q(x) = (x^2 + 1) / (x - 2). Find q(0).

These problems cover a range of function types, including quadratic functions, cubic functions, and functions involving absolute value and fractions. Take your time, carefully apply the order of operations, and show your work. For each problem, start by substituting the given input value into the function, and then follow the steps we discussed earlier to simplify the expression. Don't be afraid to make mistakes – that's how we learn! If you get stuck, go back and review the example we worked through earlier, or re-read the section on common mistakes to avoid. Once you've worked through these problems, you'll have a much better grasp of how to evaluate functions, and you'll be well on your way to mastering this important mathematical skill. Good luck, and happy calculating!

Conclusion: The Power of Function Evaluation

So, guys, we've reached the end of our journey into the world of function evaluation. We started with a specific problem – finding h(-8) for the function h(t) = -2(t+5)^2 + 4 – and we've explored not only the step-by-step solution but also the underlying concepts and potential pitfalls. We've seen how function evaluation is a fundamental skill in mathematics, with applications ranging from graphing parabolas to modeling real-world phenomena. By understanding how to plug in values, follow the order of operations, and avoid common mistakes, we can confidently tackle a wide range of function-related problems. Function evaluation is more than just plugging in numbers; it's about understanding the relationship between inputs and outputs and how functions describe that relationship. Each time we evaluate a function, we're essentially taking a snapshot of its behavior at a specific point. These snapshots, when combined, give us a complete picture of the function's overall characteristics. This understanding is crucial for anyone working with mathematical models, whether in science, engineering, economics, or any other field.

We also talked about the importance of practice. Just like any skill, function evaluation becomes easier and more natural with repeated practice. The practice problems we worked through are a great starting point, and there are countless other opportunities to hone your skills. Look for functions in your textbook, online, or even in real-world situations, and challenge yourself to evaluate them at different points. Remember, the goal isn't just to get the right answer; it's to understand the process and develop a strong conceptual foundation. The ability to evaluate functions is a powerful tool in your mathematical arsenal. It allows you to explore the behavior of functions, make predictions, and solve problems in a variety of contexts. So, keep practicing, keep exploring, and keep having fun with math! You guys got this!