Finding The Y-coordinate \[y_v\] Of The Function \[f(x) = 2x^2 - 12x\] When \[x_v = 3\]

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Hey guys! Today, we're diving into a fun math problem where we need to find the y-coordinate (yv{y_v}) of a function, given its x-coordinate (xv{x_v}). The function we're working with is f(x)=2x2−12x{f(x) = 2x^2 - 12x}, and we know that xv=3{x_v = 3}. So, let's break this down step by step to make sure we understand everything clearly. Get ready to put on your math hats!

Understanding the Problem

Before we jump into solving, let's make sure we're all on the same page. When we talk about a function like f(x)=2x2−12x{f(x) = 2x^2 - 12x}, we're describing a relationship between x{x} and y{y}. Think of it like a machine: you put in an x{x} value, and the function spits out a y{y} value. In this case, f(x){f(x)} is just another way of saying y{y}. The xv{x_v} and yv{y_v} coordinates often refer to the vertex of a parabola, which is the lowest or highest point on the curve. Knowing this helps us visualize what we're trying to find.

The Function f(x)=2x2−12x{f(x) = 2x^2 - 12x}

Let's take a closer look at our function: f(x)=2x2−12x{f(x) = 2x^2 - 12x}. This is a quadratic function, which means it forms a parabola when graphed. The shape of the parabola is determined by the coefficients of the terms. The 2x2{2x^2} part tells us that the parabola opens upwards because the coefficient 2 is positive. The −12x{-12x} part affects the position and symmetry of the parabola. Understanding these basics helps us predict the behavior of the function and makes solving for specific points much easier. It's like knowing the rules of the game before you start playing!

The Given x-coordinate: xv=3{x_v = 3}

We're given that xv=3{x_v = 3}. This is a crucial piece of information because it tells us exactly where on the x-axis we need to focus. Remember, xv{x_v} is the x-coordinate of the vertex of the parabola. The vertex is a key point because it's where the parabola changes direction. Knowing the x-coordinate of the vertex allows us to find the corresponding y-coordinate, which is what the problem asks us to do. It's like having a treasure map and knowing one of the coordinates – we're one step closer to finding the treasure!

How to Find the y-coordinate

Now that we understand the problem, let's talk about how to actually find the y-coordinate (yv{y_v}). The simplest way to do this is to substitute the given x-coordinate (xv=3{x_v = 3}) into the function f(x)=2x2−12x{f(x) = 2x^2 - 12x}. This is because the function tells us exactly how x{x} and y{y} are related. By plugging in the value of x{x}, we can calculate the corresponding value of y{y}. It's like using a recipe: if you know the ingredients and the instructions, you can bake the cake!

Step-by-step Calculation

Let's go through the calculation step by step. This will help make sure we don't miss anything and that we understand each part of the process. Here’s how we do it:

  1. Write down the function: f(x)=2x2−12x{f(x) = 2x^2 - 12x}
  2. Substitute x=3{x = 3} into the function: f(3)=2(3)2−12(3){f(3) = 2(3)^2 - 12(3)}
  3. Calculate the exponent: f(3)=2(9)−12(3){f(3) = 2(9) - 12(3)}
  4. Perform the multiplications: f(3)=18−36{f(3) = 18 - 36}
  5. Subtract to find the y-coordinate: f(3)=−18{f(3) = -18}

So, the y-coordinate (yv{y_v}) is -18. Easy peasy, right? By following these steps carefully, we can solve similar problems with confidence. It's like following a trail of breadcrumbs to find our way!

The Solution: yv=−18{y_v = -18}

We've done the math, and we've found our answer! The y-coordinate (yv{y_v}) of the function f(x)=2x2−12x{f(x) = 2x^2 - 12x} when xv=3{x_v = 3} is −18{-18}. This means that the vertex of the parabola is at the point (3, -18). This is a specific point on the graph of the function, and we've successfully located it. It's like finding the missing piece of a puzzle!

Visualizing the Result

It can be helpful to visualize what we've found. Imagine a parabola opening upwards. The vertex is the lowest point on this curve. We've determined that this lowest point is at (3, -18). This gives us a clear picture of where the function is on the coordinate plane. Visualizing the result helps us understand the problem better and makes the solution more meaningful. It's like seeing the big picture after zooming in on the details!

Why This Matters

You might be wondering, “Why is this important?” Well, finding the vertex of a parabola has many practical applications. For example, in physics, the trajectory of a projectile (like a ball thrown in the air) can be modeled by a parabola. The vertex represents the highest point the projectile reaches. In business, quadratic functions can model profit, and the vertex can represent the point of maximum profit. Understanding how to find the vertex allows us to solve real-world problems. It's like learning a new superpower that you can use in various situations!

Real-world Applications

Think about designing a bridge. Engineers need to understand the parabolic curves of suspension cables to ensure the bridge is stable and can handle weight. The vertex of the parabola helps determine the lowest point of the cable, which is crucial for structural integrity. Similarly, in sports, understanding the parabolic path of a ball can help athletes improve their performance. Whether it’s a basketball player shooting hoops or a golfer hitting a ball, the principles of parabolic motion apply. These examples show how math concepts like finding the vertex of a parabola are more than just abstract ideas – they have tangible impacts on our daily lives. It’s like seeing the gears turning in a complex machine and understanding how they all work together!

Practice Makes Perfect

Like any skill, math gets easier with practice. If you want to get better at finding coordinates of functions, try working through similar problems. Change the function, change the given x-coordinate, and see if you can find the y-coordinate. The more you practice, the more comfortable you'll become with the process. It's like learning to ride a bike: the first few tries might be wobbly, but with practice, you'll be cruising along smoothly!

Tips for Practicing

  • Start with simple functions: Begin with easier quadratic functions and gradually move to more complex ones.
  • Use graphing tools: Graphing the functions can help you visualize the parabola and the vertex, reinforcing your understanding.
  • Check your answers: Use online calculators or graphing tools to verify your solutions.
  • Work with a friend: Studying with a friend can make the process more enjoyable and help you learn from each other.

Conclusion

So, there you have it! We've successfully found the y-coordinate (yv{y_v}) of the function f(x)=2x2−12x{f(x) = 2x^2 - 12x} when xv=3{x_v = 3}. The answer is −18{-18}. We walked through the problem step by step, discussed why this matters, and even talked about how to practice. Math might seem daunting at times, but breaking it down into smaller steps can make it much more manageable. Keep practicing, keep exploring, and you'll be amazed at what you can achieve. It's like climbing a mountain: one step at a time, you'll reach the summit!

Final Thoughts

Remember, math is not just about numbers and equations; it's about problem-solving and critical thinking. The skills you learn in math can be applied to many different areas of life. So, keep challenging yourself, keep asking questions, and keep learning. You've got this! It's like building a house: each brick you lay makes the structure stronger and more resilient.

I hope this explanation was helpful and fun. Keep up the great work, guys, and I'll see you in the next math adventure!