Solving The Quadratic Function F(x) = (2/3)X^2 - (4/2)X - 7 For X = (-2, 2, 4)

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Hey guys! Today, we're diving into solving a quadratic function. Specifically, we're going to tackle the function f(x) = (2/3)X^2 - (4/2)X - 7 for the given values of X = (-2, 2, 4). Don't worry, it might look a little intimidating at first, but we'll break it down step by step so it's super easy to understand. Let's get started!

Understanding Quadratic Functions

Before we jump into solving, let's quickly recap what a quadratic function actually is. In simple terms, a quadratic function is a polynomial function of degree two. This basically means the highest power of the variable (in our case, 'X') is 2. The general form of a quadratic function is f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants. These constants determine the shape and position of the parabola, which is the graph of a quadratic function.

  • 'a': This coefficient determines whether the parabola opens upwards (if 'a' is positive) or downwards (if 'a' is negative). It also affects how "wide" or "narrow" the parabola is. A larger absolute value of 'a' makes the parabola narrower.
  • 'b': This coefficient influences the position of the parabola's axis of symmetry. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the vertex, which is the minimum or maximum point of the parabola.
  • 'c': This constant represents the y-intercept of the parabola. It's the point where the parabola intersects the y-axis.

Our function, f(x) = (2/3)X^2 - (4/2)X - 7, perfectly fits this general form. Here, a = 2/3, b = -4/2 (which simplifies to -2), and c = -7. Now that we've refreshed our understanding of quadratic functions, we can move on to solving it for the given X values.

Solving f(x) for X = -2

Okay, let's start with the first value, X = -2. To solve for f(x), we simply substitute -2 for every 'X' in our function: f(x) = (2/3)X^2 - (4/2)X - 7.

So, we get:

f(-2) = (2/3)(-2)^2 - (4/2)(-2) - 7

Now, let's break this down step-by-step:

  1. (-2)^2 = 4 (Remember, a negative number squared becomes positive).
  2. (2/3) * 4 = 8/3
  3. (-4/2) = -2
  4. (-2) * (-2) = 4

Now, let's plug these values back into our equation:

f(-2) = 8/3 + 4 - 7

To add these together, we need a common denominator. Let's convert 4 and -7 into fractions with a denominator of 3:

  • 4 = 12/3
  • -7 = -21/3

Now we can add them all up:

f(-2) = 8/3 + 12/3 - 21/3

f(-2) = (8 + 12 - 21) / 3

f(-2) = -1/3

So, when X = -2, the value of the function f(x) is -1/3. Great job, guys! We've solved it for the first value. Let's move on to the next one.

Solving f(x) for X = 2

Next up, we need to find the value of the function when X = 2. We'll follow the exact same process as before, substituting 2 for every 'X' in our function: f(x) = (2/3)X^2 - (4/2)X - 7.

So, we get:

f(2) = (2/3)(2)^2 - (4/2)(2) - 7

Let's break this down step-by-step again:

  1. (2)^2 = 4
  2. (2/3) * 4 = 8/3
  3. (-4/2) = -2
  4. (-2) * (2) = -4

Now, let's plug these values back into our equation:

f(2) = 8/3 - 4 - 7

Just like before, we need a common denominator to add these together. Let's convert -4 and -7 into fractions with a denominator of 3:

  • -4 = -12/3
  • -7 = -21/3

Now we can add them all up:

f(2) = 8/3 - 12/3 - 21/3

f(2) = (8 - 12 - 21) / 3

f(2) = -25/3

Therefore, when X = 2, the value of the function f(x) is -25/3. Awesome work, everyone! We're on a roll. Only one more value to go!

Solving f(x) for X = 4

Alright, last but not least, we need to solve for X = 4. Let's do this! We substitute 4 for every 'X' in our function: f(x) = (2/3)X^2 - (4/2)X - 7.

This gives us:

f(4) = (2/3)(4)^2 - (4/2)(4) - 7

Let's break it down one more time:

  1. (4)^2 = 16
  2. (2/3) * 16 = 32/3
  3. (-4/2) = -2
  4. (-2) * (4) = -8

Plugging these values back into our equation, we get:

f(4) = 32/3 - 8 - 7

Again, we need a common denominator to add these together. Let's convert -8 and -7 into fractions with a denominator of 3:

  • -8 = -24/3
  • -7 = -21/3

Now we add them up:

f(4) = 32/3 - 24/3 - 21/3

f(4) = (32 - 24 - 21) / 3

f(4) = -13/3

So, when X = 4, the value of the function f(x) is -13/3. Fantastic job, you guys! We've successfully solved the function for all the given X values.

Summary of Results

To recap, we've found the values of the function f(x) = (2/3)X^2 - (4/2)X - 7 for X = -2, 2, and 4:

  • f(-2) = -1/3
  • f(2) = -25/3
  • f(4) = -13/3

Visualizing the Solution

It's always helpful to visualize what we've just calculated. Imagine plotting these points on a graph. You'd see three distinct points that lie on the parabola represented by the function f(x). This gives you a visual understanding of how the function behaves for different values of X. The parabola is the u-shaped curve that represents the quadratic function. Its shape and position are determined by the coefficients and constants in the equation, which we discussed earlier.

Key Takeaways

Solving quadratic functions like this might seem tricky at first, but with a systematic approach, it becomes quite manageable. Here are some key takeaways:

  • Substitution is Key: The core of solving for specific values is substituting the given 'X' value into the function.
  • Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Common Denominators: When adding or subtracting fractions, always make sure they have a common denominator.
  • Step-by-Step Approach: Breaking down the problem into smaller, manageable steps makes it less overwhelming and reduces the chance of errors.
  • Verification: Always double-check your work! It's easy to make a small arithmetic mistake, so take a moment to review your calculations.

Conclusion

And there you have it! We've successfully solved the quadratic function f(x) = (2/3)X^2 - (4/2)X - 7 for X = -2, 2, and 4. Hopefully, this breakdown has helped you understand the process better. Remember, practice makes perfect, so try solving other quadratic functions to solidify your understanding. Keep up the great work, guys! You've got this! And most importantly, don’t be afraid to ask for help if you get stuck. There are tons of resources available, from online tutorials to math teachers who are always happy to explain things further.

Understanding these concepts not only helps in math class but also builds valuable problem-solving skills that can be applied in many areas of life. So, keep exploring, keep learning, and keep those math muscles strong! You're doing an amazing job, and remember that every problem you solve brings you one step closer to mastering mathematics. Keep challenging yourself, and you’ll be surprised at how much you can achieve!