Evaluate F(x) = 2x^2 - 2x At X = -1

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Evaluate f(x) = 2x^2 - 2x at x = -1

In this article, we're going to dive into evaluating a simple quadratic function. Specifically, we'll be finding the value of the function f(x) = 2x^2 - 2x when x = -1. This is a fundamental concept in algebra, and understanding how to do this is crucial for more advanced topics in mathematics. So, let's get started!

Understanding the Function

Before we jump into plugging in the value of x, let's take a moment to understand what the function f(x) = 2x^2 - 2x represents. This is a quadratic function, which means it's a polynomial of degree two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In our case, a = 2, b = -2, and c = 0. Quadratic functions create a parabola when graphed, which is a U-shaped curve. Understanding the basic structure of the function helps in predicting its behavior and interpreting the results we get.

Components of the Function

  • 2x^2: This term signifies that the value of x is squared and then multiplied by 2. The square operation means that the result will always be positive if x is a real number, and multiplying by 2 scales the squared value.
  • -2x: This term indicates that the value of x is multiplied by -2. This term introduces a linear component to the function, affecting the overall shape and position of the parabola.

By understanding these components, we can better appreciate how the function behaves as x changes. For example, when x is a large positive number, the 2x^2 term will dominate, making the function value large and positive. Conversely, when x is a large negative number, the 2x^2 term still dominates, making the function value large and positive, but the -2x term will contribute a positive value, moderating the overall result to some extent.

Evaluating the Function at x = -1

Now that we understand the function, let's find the value of f(x) when x = -1. This means we're going to substitute -1 for every instance of x in the function. Here's how we do it:

f(-1) = 2(-1)^2 - 2(-1)

Following the order of operations (PEMDAS/BODMAS), we first deal with the exponent:

(-1)^2 = 1

So our equation becomes:

f(-1) = 2(1) - 2(-1)

Next, we perform the multiplications:

2(1) = 2

-2(-1) = 2

Now we have:

f(-1) = 2 + 2

Finally, we do the addition:

f(-1) = 4

Therefore, the value of the function f(x) = 2x^2 - 2x when x = -1 is 4.

Step-by-Step Breakdown

  1. Substitute: Replace every x in the function with -1: f(-1) = 2(-1)^2 - 2(-1)
  2. Exponent: Evaluate (-1)^2, which equals 1: f(-1) = 2(1) - 2(-1)
  3. Multiply: Perform the multiplications: 2(1) = 2 and -2(-1) = 2
  4. Add: Add the results: f(-1) = 2 + 2 = 4

Importance of Order of Operations

It's essential to follow the order of operations (PEMDAS/BODMAS) when evaluating functions to ensure we arrive at the correct answer. The order is as follows:

  1. Parentheses / Brackets
  2. Exponents / Orders
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

If we don't follow this order, we could easily make mistakes and get the wrong result. For example, if we were to multiply 2 by -1 before evaluating the exponent in our problem, we would get a different answer. Always remember PEMDAS/BODMAS to keep your calculations accurate!

Practical Applications

Evaluating functions is a fundamental skill in mathematics and has numerous practical applications in various fields. Here are a few examples:

  • Physics: In physics, functions are used to model the motion of objects. For example, the position of a projectile can be described as a function of time. Evaluating this function at a specific time gives you the position of the projectile at that moment.
  • Engineering: Engineers use functions to design and analyze structures, circuits, and systems. For example, the stress on a beam can be described as a function of the load applied. Evaluating this function helps engineers ensure that the beam can withstand the load without failing.
  • Economics: Economists use functions to model economic phenomena such as supply and demand. For example, the quantity of a product demanded by consumers can be described as a function of the price. Evaluating this function helps economists understand how changes in price affect demand.
  • Computer Science: In computer science, functions are used extensively in programming. They allow programmers to break down complex tasks into smaller, more manageable units. Evaluating a function means running the code within the function with specific inputs to produce an output.

Common Mistakes to Avoid

When evaluating functions, it's easy to make mistakes, especially if you're just starting out. Here are some common mistakes to watch out for:

  • Incorrect Substitution: Make sure you replace every instance of x with the correct value. It's easy to miss one, especially in more complex functions.
  • Order of Operations Errors: As mentioned earlier, always follow the order of operations (PEMDAS/BODMAS). Failing to do so can lead to incorrect results.
  • Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. A simple sign error can throw off your entire calculation.
  • Arithmetic Errors: Double-check your arithmetic. Simple addition, subtraction, multiplication, or division errors can happen if you're not careful.

To avoid these mistakes, take your time, write out each step clearly, and double-check your work. Practice makes perfect!

Conclusion

In this article, we've walked through the process of evaluating the function f(x) = 2x^2 - 2x at x = -1. We found that f(-1) = 4. We also discussed the importance of understanding the function, following the order of operations, and avoiding common mistakes. Evaluating functions is a fundamental skill in mathematics with wide-ranging applications, so mastering it is essential for success in many fields. Keep practicing, and you'll become a pro in no time!