Solving For F(2+1) - F(-2-1) With F(x) = 4x^2 - 8x + 3

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Hey guys! Let's dive into this math problem where we're given a function and asked to find the difference between the function evaluated at two different points. It might seem a bit daunting at first, but trust me, we'll break it down step by step and you'll see it's totally manageable.

Understanding the Function

First things first, let's understand the function we're working with: f(x) = 4x^2 - 8x + 3. This is a quadratic function, which means it has an x-squared term. Knowing this helps us anticipate the shape of the graph (a parabola) and the types of calculations we'll be doing. The key here is to remember that f(x) simply means we're taking an input value 'x', plugging it into the expression, and getting an output value.

Breaking Down the Problem

Our main goal is to find the value of f(2 + 1) - f(-2 - 1). Notice how we have two parts here: f(2 + 1) and f(-2 - 1). We need to evaluate each of these separately before we can subtract them. This is a classic example of breaking a complex problem into smaller, more manageable pieces. First, let's simplify the expressions inside the parentheses. 2 + 1 is simply 3, and -2 - 1 is -3. So, we're actually trying to find f(3) - f(-3). Now it looks a lot less scary, right?

Calculating f(3)

To calculate f(3), we'll substitute x with 3 in our function: f(3) = 4(3)^2 - 8(3) + 3. Remember the order of operations (PEMDAS/BODMAS)? We need to do the exponent first: 3 squared (3^2) is 9. So, now we have: f(3) = 4(9) - 8(3) + 3. Next up, multiplication: 4 times 9 is 36, and 8 times 3 is 24. Our expression now looks like this: f(3) = 36 - 24 + 3. Finally, we do the addition and subtraction from left to right: 36 - 24 is 12, and 12 + 3 is 15. So, we've found that f(3) = 15.

Calculating f(-3)

Now, let's tackle f(-3). We'll do the same thing, but this time we're substituting x with -3: f(-3) = 4(-3)^2 - 8(-3) + 3. Again, we start with the exponent: (-3) squared is (-3) * (-3), which equals 9. Remember, a negative times a negative is a positive! So, we have: f(-3) = 4(9) - 8(-3) + 3. Next, the multiplication: 4 times 9 is 36, and -8 times -3 is +24 (again, negative times negative is positive). Now we have: f(-3) = 36 + 24 + 3. Adding these up, 36 + 24 is 60, and 60 + 3 is 63. Therefore, f(-3) = 63.

Finding the Difference

We're almost there! We've calculated f(3) and f(-3). Now we just need to find the difference: f(3) - f(-3). We know that f(3) = 15 and f(-3) = 63, so we're doing 15 - 63. This gives us -48. So, the final answer is f(2 + 1) - f(-2 - 1) = -48.

Why This Matters

Understanding how to evaluate functions is a fundamental skill in mathematics. It's not just about plugging in numbers; it's about understanding the relationship between inputs and outputs. Functions are used everywhere, from physics and engineering to economics and computer science. Being comfortable with them opens doors to understanding more complex concepts later on. Plus, the step-by-step approach we used here – breaking down a problem, handling each part carefully, and then putting it all together – is a valuable strategy for solving all sorts of problems, not just math ones.

Practice Makes Perfect

The best way to really master this is to practice. Try different functions and different input values. Play around with it! You can even make up your own functions and challenge yourself or your friends. The more you practice, the more natural this will become.

Tips for Success

  • Write it out: Don't try to do everything in your head. Write down each step clearly. This helps you keep track of what you're doing and reduces the chance of making mistakes.
  • Order of operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This is the key to getting the correct answer.
  • Double-check: Once you have an answer, go back and check your work. Did you make any simple arithmetic errors? Did you follow the order of operations correctly?
  • Don't be afraid to ask for help: If you're stuck, don't spin your wheels. Ask a teacher, a friend, or look for resources online. There's no shame in needing a little help, and often a fresh perspective is all you need.

Conclusion

So, there you have it! We've successfully found the value of f(2 + 1) - f(-2 - 1) for the given function. Remember, the key is to break the problem down, evaluate each part carefully, and then combine the results. Keep practicing, and you'll be a function-evaluating pro in no time! Math can be challenging, but it's also incredibly rewarding when you understand it. Keep up the great work, guys!

Hey there, math enthusiasts! Today, we're going to dissect a fascinating function, f(x) = 4x^2 - 8x + 3, and explore how to work with it effectively. This function isn't just a random equation; it's a gateway to understanding a whole world of mathematical concepts. We'll not only solve specific problems, but also delve into the broader implications and applications of functions like this. Think of this as your ultimate guide to conquering quadratic functions!

What is a Function, Anyway?

Before we jump into the specifics of f(x) = 4x^2 - 8x + 3, let's take a step back and define what a function actually is. Simply put, a function is a rule that assigns each input value (x) to exactly one output value (f(x)). Think of it like a machine: you put something in (the input), the machine does something to it, and something else comes out (the output). In our case, the “machine” is the equation f(x) = 4x^2 - 8x + 3. We input a value for 'x', and the function spits out a corresponding value for 'f(x)'.

Functions are the cornerstone of mathematics and are used to model relationships between different quantities in the real world. From the trajectory of a ball thrown in the air to the growth of a population, functions help us understand and predict how things behave. Mastering functions is crucial for success in higher-level math courses and many STEM fields.

Diving Deep into f(x) = 4x^2 - 8x + 3

Now, let's get back to our specific function: f(x) = 4x^2 - 8x + 3. As we mentioned earlier, this is a quadratic function because it has an x^2 term. Quadratic functions have a few key characteristics that are worth noting:

  • Parabola: The graph of a quadratic function is a parabola, a U-shaped curve. This shape arises because the x^2 term causes the function to increase or decrease more rapidly as x moves away from the vertex (the turning point of the parabola).
  • Vertex: The vertex is the minimum (or maximum) point of the parabola. It's a crucial point because it tells us the lowest (or highest) value the function will reach.
  • Roots (or Zeros): The roots are the x-values where the function crosses the x-axis (i.e., where f(x) = 0). Finding the roots is a common problem in algebra and has many practical applications.

Understanding these characteristics helps us visualize the function and anticipate its behavior. For example, we know that f(x) = 4x^2 - 8x + 3 will form a parabola, and we can use this knowledge to solve problems and make predictions.

Evaluating the Function: Plugging in Values

One of the most basic things we can do with a function is to evaluate it at a specific value of x. This simply means substituting the value into the equation and calculating the result. We did this earlier when we found f(3) and f(-3). Let's try a few more examples to solidify this skill.

  • Example 1: Find f(0)

    To find f(0), we replace every 'x' in the equation with '0':

    f(0) = 4(0)^2 - 8(0) + 3

    f(0) = 4(0) - 8(0) + 3

    f(0) = 0 - 0 + 3

    f(0) = 3

  • Example 2: Find f(1)

    Similarly, to find f(1), we replace 'x' with '1':

    f(1) = 4(1)^2 - 8(1) + 3

    f(1) = 4(1) - 8(1) + 3

    f(1) = 4 - 8 + 3

    f(1) = -1

  • Example 3: Find f(-1)

    And for f(-1):

    f(-1) = 4(-1)^2 - 8(-1) + 3

    f(-1) = 4(1) + 8 + 3

    f(-1) = 4 + 8 + 3

    f(-1) = 15

Notice how the output value f(x) changes depending on the input value x. This is the essence of a function – it maps different inputs to different outputs.

Finding the Vertex: Unveiling the Turning Point

The vertex of a quadratic function is a crucial point. It represents the minimum or maximum value of the function. For our function, f(x) = 4x^2 - 8x + 3, the coefficient of the x^2 term (which is 4) is positive. This means the parabola opens upwards, and the vertex will be the minimum point.

There are a couple of ways to find the vertex:

  1. Completing the Square: This method involves rewriting the quadratic function in vertex form: f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Completing the square is a powerful technique, but it can be a bit involved.
  2. Using the Vertex Formula: This is a shortcut that gives us the x-coordinate of the vertex directly: h = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. Once we have the x-coordinate (h), we can plug it back into the function to find the y-coordinate (k).

Let's use the vertex formula for our function. In f(x) = 4x^2 - 8x + 3, a = 4 and b = -8. So,

h = -(-8) / (2 * 4) = 8 / 8 = 1

The x-coordinate of the vertex is 1. Now, let's find the y-coordinate by plugging h = 1 back into the function:

k = f(1) = 4(1)^2 - 8(1) + 3 = 4 - 8 + 3 = -1

Therefore, the vertex of the parabola is (1, -1). This tells us that the minimum value of the function is -1, and it occurs when x = 1.

Finding the Roots: Where the Function Crosses the Axis

The roots (or zeros) of a quadratic function are the x-values where the function equals zero, i.e., where the parabola crosses the x-axis. Finding the roots is a fundamental problem in algebra with many applications, such as solving equations and modeling physical phenomena.

There are several ways to find the roots:

  1. Factoring: If the quadratic expression can be factored, we can set each factor equal to zero and solve for x. This is the quickest method, but it doesn't always work.
  2. Quadratic Formula: This formula works for any quadratic equation, regardless of whether it can be factored. The quadratic formula is: x = [-b ± √(b^2 - 4ac)] / 2a, where a, b, and c are the coefficients of the quadratic equation.
  3. Completing the Square: We can also find the roots by completing the square and then solving for x.

Let's use the quadratic formula to find the roots of our function, f(x) = 4x^2 - 8x + 3. In this case, a = 4, b = -8, and c = 3. Plugging these values into the quadratic formula, we get:

x = [8 ± √((-8)^2 - 4 * 4 * 3)] / (2 * 4)

x = [8 ± √(64 - 48)] / 8

x = [8 ± √16] / 8

x = [8 ± 4] / 8

Now we have two possible solutions:

x1 = (8 + 4) / 8 = 12 / 8 = 3/2

x2 = (8 - 4) / 8 = 4 / 8 = 1/2

So, the roots of the function are x = 3/2 and x = 1/2. This means the parabola crosses the x-axis at these two points.

Applications of Quadratic Functions: Real-World Examples

Quadratic functions aren't just abstract mathematical concepts; they have numerous applications in the real world. Here are a few examples:

  • Projectile Motion: The path of a projectile (like a ball thrown in the air) can be modeled by a quadratic function. The vertex represents the maximum height reached by the projectile, and the roots can tell us when the projectile hits the ground.
  • Optimization Problems: Quadratic functions are often used in optimization problems, where we want to find the maximum or minimum value of a certain quantity. For example, we might use a quadratic function to find the dimensions of a rectangular garden that maximize its area for a given perimeter.
  • Engineering: Quadratic functions are used in engineering to design bridges, buildings, and other structures. They help engineers calculate stresses, strains, and other important factors.
  • Economics: Quadratic functions can be used to model cost, revenue, and profit functions in economics. This helps businesses make decisions about pricing, production, and other factors.

These are just a few examples, but they illustrate the wide range of applications of quadratic functions. Understanding these functions is essential for anyone pursuing a career in math, science, engineering, or related fields.

Going Beyond: Further Exploration of Functions

We've covered a lot of ground in this discussion, but there's much more to explore when it comes to functions. Here are a few ideas for further learning:

  • Other Types of Functions: Explore linear, exponential, logarithmic, trigonometric, and other types of functions. Each type has its own unique properties and applications.
  • Transformations of Functions: Learn how to shift, stretch, and reflect graphs of functions. This can give you a deeper understanding of how functions behave.
  • Calculus: Calculus is the study of change, and functions are the building blocks of calculus. Learning calculus will allow you to analyze functions in even greater detail.
  • Real-World Projects: Look for real-world projects that involve functions. This will help you see the practical applications of the concepts you're learning.

Conclusion: Your Journey with Functions is Just Beginning

Congratulations! You've taken a significant step in understanding functions, particularly the quadratic function f(x) = 4x^2 - 8x + 3. We've covered the basics of what a function is, how to evaluate it, how to find its vertex and roots, and how it's used in the real world. But remember, this is just the beginning. The world of functions is vast and fascinating, and there's always more to learn. Keep exploring, keep practicing, and you'll be amazed at what you can discover.

Functions are the language of mathematics, and mastering them will open doors to a deeper understanding of the world around you. So, keep up the great work, and never stop learning!