Evaluating H(t) = 2t^2 + 9 At T = 5: A Step-by-Step Guide

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Hey guys! Today, we're diving into the world of functions and learning how to evaluate them. Specifically, we'll be looking at the function h(t) = 2t^2 + 9 and figuring out what its value is when t = 5. Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step, so you'll be a pro in no time.

Understanding Functions

Before we jump into the evaluation, let's quickly recap what a function actually is. Think of a function like a machine: you feed it an input, it does some calculations, and then it spits out an output. In our case, the function h(t) = 2t^2 + 9 takes an input value, which we call t, performs some operations on it (squaring, multiplying by 2, and adding 9), and then gives us the result, which is the value of h(t). This is a fundamental concept in mathematics, and grasping it is crucial for tackling more complex problems later on. The beauty of functions lies in their ability to model real-world relationships. For instance, the height of a ball thrown in the air can be represented as a function of time. The cost of manufacturing a product can be represented as a function of the number of units produced. Understanding functions allows us to make predictions and solve problems in a variety of fields, from physics and engineering to economics and computer science. So, let's solidify our understanding by working through this example together!

The Task at Hand: Evaluating h(5)

Our mission today is to evaluate h(5). What does this mean? Well, it simply means we need to find the value of the function h(t) when t is equal to 5. We're essentially plugging in 5 for t in the function's equation and then simplifying. This process of substitution is key to evaluating functions. It's like giving our function machine the input value of 5 and waiting to see what output it produces. Evaluating functions is a core skill in algebra and calculus, and it's used extensively in various applications. For example, in physics, you might need to evaluate a function that represents the position of an object at a specific time. In economics, you might need to evaluate a cost function to determine the cost of producing a certain number of goods. Mastering this skill will undoubtedly open doors to solving a wider range of problems. Now, let's roll up our sleeves and get to the actual calculation!

Step-by-Step Solution

Okay, let's get our hands dirty and evaluate h(5) step-by-step:

  1. Write down the function: Our function is h(t) = 2t^2 + 9. This is our starting point, our recipe for the calculation. It's crucial to have the correct function written down before we proceed; a small error here can throw off the entire result. So, double-check that you've got the equation right.

  2. Substitute t with 5: This is the core of the evaluation process. Wherever we see a t in the function, we replace it with 5. So, h(t) = 2t^2 + 9 becomes h(5) = 2(5)^2 + 9. Make sure you're careful with the parentheses! They ensure that we square the 5 before multiplying by 2. This substitution step is fundamental to understanding function evaluation. It's the act of feeding the input value into the function's formula. We're essentially saying, "Okay, function, here's the input 5; what do you do with it?"

  3. Simplify: Now comes the arithmetic! We need to follow the order of operations (PEMDAS/BODMAS) to simplify the expression. First, we handle the exponent: (5)^2 = 25. So, our equation becomes h(5) = 2(25) + 9. Next, we perform the multiplication: 2(25) = 50. This gives us h(5) = 50 + 9. Finally, we do the addition: 50 + 9 = 59. This simplification process is where we actually calculate the output of the function. We're taking the expression we got after substitution and working through the mathematical operations to arrive at a single numerical value. It's like the "cooking" stage of our recipe, where we combine the ingredients (the numbers and operations) to create the final dish (the function's value).

  4. The result: Therefore, h(5) = 59. This is our final answer! We've successfully evaluated the function h(t) at t = 5. This answer represents the output of our function machine when we input 5. It's the specific value that the function takes on at that particular point. This result can be interpreted in different ways depending on the context of the problem. For instance, if h(t) represented the height of an object after t seconds, then h(5) = 59 would tell us that the object is 59 units high after 5 seconds. This is the power of function evaluation – it allows us to obtain concrete numerical values from abstract mathematical expressions.

Common Mistakes to Avoid

Evaluating functions is pretty straightforward, but there are a few common pitfalls you might encounter. Let's look at some of these so you can steer clear:

  • Order of Operations: This is a big one! Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Make sure you're squaring before multiplying, multiplying before adding, and so on. A slip-up in the order of operations can completely change your answer. For example, if you multiply 2 by 5 before squaring, you'll get a very different result. Sticking to the correct order is paramount for accurate calculations.

  • Incorrect Substitution: Double-check that you're replacing every instance of t with 5 (or whatever value you're evaluating at). Don't miss any! Sometimes, when a function has multiple occurrences of the variable, it's easy to overlook one. A careful, systematic substitution is crucial to avoid errors. It's a good idea to visually scan the function after substitution to ensure that all the t's have been replaced.

  • Sign Errors: Be careful with negative signs! Squaring a negative number results in a positive number, but a single missed negative can throw off your entire calculation. Pay close attention to the signs of each term as you simplify the expression. Remember the rules of sign multiplication (negative times negative is positive, negative times positive is negative) and apply them consistently.

  • Misunderstanding the Notation: Remember that h(5) means "the value of the function h when t is 5," not "h multiplied by 5." This is a crucial distinction. Function notation is a shorthand way of representing the output of a function for a given input. It's not a multiplication operation. Mixing up function notation with multiplication is a common mistake, so make sure you understand what the notation truly signifies.

Practice Makes Perfect

The best way to master function evaluation is to practice, practice, practice! Try evaluating h(t) at different values of t, like 0, -2, or even fractions. You can also try evaluating other functions, like f(x) = x^3 - 4x + 1 or g(y) = (y + 3) / (y - 1). The more you practice, the more comfortable you'll become with the process.

Evaluating functions is a fundamental skill in mathematics, and it's essential for understanding more advanced concepts. By following these steps and avoiding common mistakes, you'll be evaluating functions like a pro in no time! So go ahead, grab some practice problems, and get started. You've got this!