Evaluate H(t) = T^2 + 2t + 1 At T = -2: A Step-by-Step Guide

Hey guys! Let's dive into a fun little math problem today. We're given a function, a mathematical expression that does something cool with a variable (in this case, the variable is 't'). The function is defined as h(t) = t² + 2t + 1. Our mission, should we choose to accept it (and we totally do!), is to figure out what happens when we plug in -2 for 't'. In other words, we need to find the value of h(-2). Sounds like a blast, right? Let's get started!

Understanding the Function

Before we jump straight into the calculation, let's take a moment to appreciate this function, h(t) = t² + 2t + 1. It's a quadratic function, which means it has a 't²' term. Quadratic functions are super important in math and science because they describe curves called parabolas. You might have seen parabolas before – they look like a U-shape, and they show up in all sorts of places, like the path of a ball thrown through the air or the shape of a satellite dish. Understanding the structure of our function helps us predict its behavior and makes solving problems like this much easier.

This particular quadratic function is written in a standard form: at² + bt + c, where 'a', 'b', and 'c' are constants (just regular numbers). In our case, a = 1, b = 2, and c = 1. These constants determine the shape and position of the parabola. Notice anything special about these coefficients? If we are sharp-eyed math detectives, we might spot that this quadratic expression is actually a perfect square! That's right, t² + 2t + 1 can be factored into (t + 1)². This is going to make our calculation even simpler later on, but let's not get ahead of ourselves. For now, just keep in mind that this function has a special structure.

Knowing that h(t) represents a parabola, we can visualize what we're doing. We're essentially asking, “What is the height of the parabola when t = -2?” The value of h(-2) will give us the y-coordinate of a point on the parabola where the x-coordinate (or the t-coordinate in this case) is -2. Visualizing functions like this is a powerful tool in mathematics, and it helps build a deeper understanding beyond just crunching numbers. So, with a solid understanding of what our function represents, let's move on to the calculation!

Step-by-Step Calculation of h(-2)

Okay, let's get down to business and actually calculate h(-2). The process is straightforward: we replace every 't' in the function's expression with '-2' and then simplify using the order of operations (PEMDAS/BODMAS). Remember that? Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following the order of operations ensures we get the correct answer.

Here’s how it looks:

h(-2) = (-2)² + 2(-2) + 1

First, let's deal with the exponent. (-2)² means -2 multiplied by itself, which is (-2) * (-2) = 4. Remember that multiplying two negative numbers results in a positive number. So, our expression now becomes:

h(-2) = 4 + 2(-2) + 1

Next up is the multiplication. We have 2 multiplied by -2, which is -4. Now our expression looks like this:

h(-2) = 4 + (-4) + 1

Now we just have addition left. Adding a negative number is the same as subtracting, so we can rewrite 4 + (-4) as 4 - 4, which equals 0. Our expression simplifies to:

h(-2) = 0 + 1

Finally, 0 + 1 is simply 1. Therefore:

h(-2) = 1

And there we have it! We've successfully calculated the value of the function h(t) when t = -2. The result is 1. Easy peasy, right? We took it one step at a time, and now we've got our answer. But, like any good mathematicians, let’s check our work and see if there's another way to arrive at the same result.

Verification and Alternative Approach

It's always a good idea to double-check our work, especially in math. We want to be absolutely sure we've got the correct answer. One way to verify our result is to go back through our steps and make sure we didn't make any arithmetic errors. Another, even cooler, method is to use our earlier observation that h(t) = t² + 2t + 1 is actually a perfect square. We realized that it can be factored into (t + 1)². This gives us an alternative way to calculate h(-2).

Let's rewrite our function as h(t) = (t + 1)². Now, to find h(-2), we substitute -2 for 't' in this new form:

h(-2) = (-2 + 1)²

First, we deal with the parentheses. -2 + 1 equals -1. So, we have:

h(-2) = (-1)²

Then, we square -1, which means multiplying -1 by itself: (-1) * (-1) = 1. Therefore:

h(-2) = 1

Guess what? We got the same answer! This confirms that our initial calculation was indeed correct. Using this alternative approach not only verifies our result but also highlights the beauty of mathematics – often there are multiple paths to the same solution. Recognizing patterns and simplifying expressions, like we did by factoring the perfect square, can make calculations easier and provide valuable insights.

Conclusion: h(-2) = 1

So, to recap, we were given the function h(t) = t² + 2t + 1 and tasked with finding the value of h(-2). We carefully substituted -2 for 't' in the function and, following the order of operations, arrived at the answer: h(-2) = 1. We then verified our result by recognizing that the function could be rewritten as h(t) = (t + 1)², which led us to the same solution.

This exercise demonstrates a fundamental concept in mathematics: evaluating functions. We've learned how to plug in a specific value for a variable and calculate the corresponding output of the function. This skill is crucial for understanding and working with mathematical models in various fields, from physics and engineering to economics and computer science.

More than just finding the answer, we've also explored the structure of the function and discovered its connection to parabolas and perfect squares. This deeper understanding enhances our mathematical intuition and problem-solving abilities. So, next time you encounter a function, remember to not just crunch the numbers but also try to understand its underlying properties and how different approaches can lead to the same result. Keep practicing, and you'll become a function-evaluating pro in no time! Great job, everyone!