Finding Y(10z) For Y(z) = Z^3 - 11z^2: A Step-by-Step Guide

Hey guys! Let's dive into a fun math problem today. We're going to figure out how to find y(10z) when we know that y(z) = z^3 - 11z^2. This might seem a bit abstract at first, but don't worry, we'll break it down step by step so it's super clear. We'll cover the basics of function evaluation, walk through the substitution process, and simplify the expression to get our final answer. So, grab your thinking caps, and let's get started!

Understanding Function Evaluation

Before we jump into the problem, let's quickly recap what function evaluation is all about. In simple terms, a function is like a machine that takes an input, does something to it, and spits out an output. We usually write a function like this: y(z) = z^3 - 11z^2. Here, y is the name of the function, and z is the input variable. The expression z^3 - 11z^2 tells us what the function does to the input.

So, when we see something like y(2), it means we're plugging in the value 2 for z in the expression. In this case, y(2) = (2)^3 - 11(2)^2 = 8 - 44 = -36. That's the basic idea! We're just replacing the variable with a specific value and then simplifying.

Now, the key thing to remember here is that whatever is inside the parentheses on the left side of the equation, we substitute it for the variable on the right side. This might sound obvious, but it’s crucial for tackling problems like finding y(10z). The beauty of function evaluation lies in its consistent rule: replace the variable! Think of it as a simple find-and-replace operation, but in the world of math.

When we're dealing with more complex inputs, like expressions involving variables (in our case, 10z), the same principle applies. We simply substitute the entire expression 10z wherever we see z in the function's definition. This is where things might seem a little tricky at first, but with practice, it becomes second nature. Understanding this substitution process is fundamental not just for this problem, but for many areas of mathematics and beyond. So, make sure you've got this concept down pat before moving on! We'll use this knowledge to tackle the core of our problem, which is finding y(10z). Remember, it's all about replacing the variable z with the given input, which in this case is 10z. This straightforward substitution is the key to unlocking the solution.

Substituting 10z into the Function

Okay, now for the fun part! We need to find y(10z), and we know that y(z) = z^3 - 11z^2. So, what do we do? We substitute 10z for every z in the equation. This means we replace z with (10z) wherever we see it. So, the equation becomes:

y(10z) = (10z)^3 - 11(10z)^2

See? It's just a direct replacement. We've swapped out the z for 10z. Now, we've got a new expression, and the next step is to simplify it. This is where our algebra skills come into play. We need to remember the order of operations (PEMDAS/BODMAS) and how to handle exponents correctly.

The first thing we'll focus on is the exponents. We have (10z)^3 and (10z)^2. Remember that when we raise a product to a power, we raise each factor to that power. So, (10z)^3 is the same as 10^3 * z^3, and (10z)^2 is the same as 10^2 * z^2. This is a crucial step because it allows us to separate the numerical coefficients from the variable terms, making the simplification process much clearer.

Next, we calculate the powers of 10. We know that 10^3 = 1000 and 10^2 = 100. Plugging these values back into our expression, we get:

y(10z) = 1000z^3 - 11(100z^2)

We're getting closer! Now, we just have one more multiplication to take care of before we have our simplified expression. Notice how each step builds upon the previous one. By carefully applying the rules of algebra and breaking down the problem into smaller, manageable steps, we're making steady progress towards the solution. Don't rush through this process; it's important to understand each step to avoid making mistakes. The goal is not just to get the right answer, but to understand why we're doing what we're doing.

Simplifying the Expression

Alright, let's finish this up! We're at the stage where we have: y(10z) = 1000z^3 - 11(100z^2). The only thing left to do is multiply the 11 by the 100z^2 term. This is a straightforward multiplication:

11 * 100z^2 = 1100z^2

So, now we can substitute this back into our equation:

y(10z) = 1000z^3 - 1100z^2

And there you have it! We've simplified the expression as much as we can. There are no more like terms to combine, and we've handled all the exponents and multiplications. This is our final answer for y(10z).

It's important to recognize that simplification is a crucial part of many math problems. Often, the initial expression you get after substitution might look complicated, but with a few steps of simplification, it can become much cleaner and easier to understand. In this case, we went from a somewhat intimidating expression with parentheses and exponents to a simple polynomial with two terms. This final form is not only easier to write but also easier to work with if we needed to use this result in further calculations.

Furthermore, simplifying expressions helps us to see the structure of the mathematical relationship more clearly. In our final expression, y(10z) = 1000z^3 - 1100z^2, we can easily see the coefficients and the powers of z. This can be useful for understanding the behavior of the function or for comparing it to other functions. So, remember that simplification is not just about getting to a final answer; it's also about gaining a deeper understanding of the math involved. Pat yourselves on the back, guys! You've successfully navigated through function evaluation, substitution, and simplification. You're becoming math whizzes!

Final Answer

So, to recap, we started with the function y(z) = z^3 - 11z^2 and the question of finding y(10z). We understood the concept of function evaluation, where we substitute a given value or expression for the variable in the function. We then carefully substituted 10z for z in the function, resulting in the expression y(10z) = (10z)^3 - 11(10z)^2. Finally, we simplified this expression step by step, using the rules of exponents and order of operations, to arrive at our final answer:

y(10z) = 1000z^3 - 1100z^2

This is the simplified form of y(10z) for the given function y(z). We've successfully solved the problem! Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. Don't try to do everything at once. Focus on one step at a time, and you'll find that even complex problems become much easier to handle. And most importantly, practice makes perfect! The more you work with function evaluation and simplification, the more comfortable you'll become with these concepts.

Now that you've mastered this problem, you can try similar ones with different functions and different substitutions. Experiment with different expressions inside the parentheses, and see how the simplification process changes. You can also explore how these types of function evaluations are used in real-world applications, such as in physics, engineering, and computer science. Math is all around us, and understanding these fundamental concepts can open up a whole new world of possibilities.

And that's it for this math adventure! I hope you found this explanation helpful and that you're now feeling confident in your ability to tackle similar problems. Remember to keep practicing, keep exploring, and most importantly, keep having fun with math! You guys are doing great!