Finding Y For F(x) = -3x^2 + 20: A Step-by-Step Guide

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Finding y for f(x) = -3x^2 + 20: A Step-by-Step Guide

Hey guys! Let's dive into a common math problem: figuring out the value of y when we're given a function like f(x) = -3x² + 20. This might seem tricky at first, but trust me, it's totally manageable once you break it down. We're going to go through this step-by-step, making sure you understand the logic behind each move. So, grab your thinking caps, and let's get started!

Understanding the Function: f(x) = -3x² + 20

Okay, first things first, let's break down what this function, f(x) = -3x² + 20, actually means. In math terms, this is a quadratic function. But don't let that fancy term scare you! Think of it as a machine. You feed it a value for x, and it spits out a corresponding value for y. The function tells you exactly how to process x to get y.

  • f(x): This is just a fancy way of saying "y". So, whenever you see f(x), you can mentally replace it with y. They are the same thing.
  • -3x²: This is the heart of the quadratic part. It means you take your x value, square it (multiply it by itself), and then multiply the result by -3. The negative sign flips the parabola downwards and the 3 stretches it, making it narrower. Understanding the coefficient -3 is crucial because it dictates the direction and shape of the parabola we'd graph from this equation. A negative coefficient means the parabola opens downwards, and the larger the absolute value of the coefficient, the narrower the parabola becomes. So, in our case, the parabola opens downwards and is stretched vertically compared to the basic x² parabola.
  • + 20: This is a constant term. It simply means that whatever you get from the -3x² part, you add 20 to it. This constant term shifts the entire parabola vertically. In our case, the +20 shifts the parabola 20 units upwards on the y-axis. This is the y-intercept of the graph, meaning it's where the parabola crosses the y-axis. This is super important because it defines a key characteristic of the function's graph, allowing us to quickly visualize its vertical position.

So, in a nutshell, the function takes an x value, squares it, multiplies by -3, and then adds 20 to give you the y value. This might seem like a lot, but with practice, it becomes second nature!

Finding y: The Process

Now that we understand what the function does, let's talk about how to actually find the value of y. The key thing to remember is that you need a value for x to plug into the function. Without an x value, you can't calculate y. Let's explore the two main scenarios you might encounter:

Scenario 1: Given a Specific x Value

This is the most straightforward case. You're given a specific value for x, and all you need to do is substitute that value into the function and do the math. This is where understanding order of operations becomes key! Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction)? We need to follow that order meticulously to arrive at the correct value for y. It's super easy to make a small arithmetic mistake if you rush through this, so take your time and double-check your work!

Let's walk through an example. Suppose we're given x = 2. To find y, we'll follow these steps:

  1. Substitute: Replace x in the function with 2: f(2) = -3(2)² + 20
  2. Exponents: Calculate 2 squared: f(2) = -3(4) + 20 This step is crucial. We need to square the 2 before we multiply by -3. Mess this up and the whole answer is off!
  3. Multiplication: Multiply -3 by 4: f(2) = -12 + 20 Remember the rules for multiplying negative and positive numbers! A negative times a positive is a negative. It's a common mistake to forget the negative sign, so keep an eye out for that.
  4. Addition: Add -12 and 20: f(2) = 8

So, when x = 2, y = 8. Easy peasy!

Let's try another one. What if x = -1?

  1. Substitute: f(-1) = -3(-1)² + 20
  2. Exponents: Calculate (-1) squared: f(-1) = -3(1) + 20 Remember, a negative number squared is positive! This is another common spot for mistakes. Always double-check the signs.
  3. Multiplication: Multiply -3 by 1: f(-1) = -3 + 20
  4. Addition: Add -3 and 20: f(-1) = 17

Therefore, when x = -1, y = 17.

See? It's all about plugging in the x value and carefully following the order of operations. Practice makes perfect, so try a few more examples on your own!

Scenario 2: Solving for x Given a y Value

This is a slightly more challenging, but still totally doable. Sometimes, instead of being given x, you're given a y value (or f(x) value) and you need to work backward to find the x value(s) that produce that y. This involves solving an equation.

Let's say we're given y = 5. To find the x values that make f(x) = 5, we need to set up the equation:

-3x² + 20 = 5

Now, we need to isolate x². This involves undoing the operations that are being done to x, but in reverse order. Think of it like peeling an onion – you have to remove the outer layers first to get to the center.

  1. Subtract 20 from both sides: This gets rid of the +20 on the left side: -3x² = -15

    Remember, whatever you do to one side of the equation, you have to do to the other to keep it balanced! It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level.

  2. Divide both sides by -3: This isolates the x² term: x² = 5

  3. Take the square root of both sides: This is the key step to get x by itself. But here's a crucial thing to remember: when you take the square root, there are always two possible solutions – a positive one and a negative one! x = ±√5

    So, in this case, we have two solutions: x = √5 and x = -√5. These are the two x values that will give you y = 5 when plugged into the function.

This step of remembering both the positive and negative square roots is super important and a common source of errors. Always double-check to see if both solutions make sense in the context of the problem!

Let's try another example. What if y = 20?

  1. Set up the equation: -3x² + 20 = 20
  2. Subtract 20 from both sides: -3x² = 0
  3. Divide both sides by -3: x² = 0
  4. Take the square root of both sides: x = 0

In this case, there's only one solution: x = 0. This makes sense, because the +20 in the original function shifts the entire parabola upwards by 20 units, meaning the vertex (the highest point in this case, since it opens downwards) is at y = 20. So, only x = 0 will give you y = 20.

Key Takeaways and Tips

  • f(x) is just another way of writing y. Don't let the notation confuse you.
  • Follow the order of operations (PEMDAS/BODMAS). This is crucial for accurate calculations.
  • When solving for x, remember the ± when taking the square root. There are usually two solutions.
  • Practice makes perfect! The more you work through these types of problems, the easier they become.

Practice Problems

To really nail this down, let's try a few practice problems.

  1. Given f(x) = -3x² + 20, find y when x = 0.
  2. Given f(x) = -3x² + 20, find y when x = 3.
  3. Given f(x) = -3x² + 20, find x when y = -7.

Work through these problems, and then check your answers. If you get stuck, go back and review the steps we covered earlier.

Conclusion

So, there you have it! Finding the value of y for a function like f(x) = -3x² + 20 is all about understanding the function and carefully following the steps. Whether you're given x or y, the key is to break the problem down into smaller, manageable chunks. With a little practice, you'll be solving these problems like a pro! Remember, math is like building with Lego bricks - each concept builds upon the previous one. Master this, and you'll be well on your way to tackling more complex problems. Keep practicing, guys, and you'll get there! If you have any questions, don't hesitate to ask. Happy calculating!