Finding The Maximum Value Of Quadratic Function F(x) = 12 + 4x - 3x^2

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In the realm of mathematics, quadratic functions hold a special place. These functions, characterized by their parabolic curves, frequently appear in various applications, from physics to engineering. Understanding how to determine the maximum or minimum value of a quadratic function is a fundamental skill. In this article, we will delve into the process of finding the maximum value of a given quadratic function, complete with a step-by-step explanation and practical insights.

Understanding Quadratic Functions

Before we embark on the journey of finding the maximum value, let's first grasp the essence of quadratic functions. A quadratic function is a polynomial function of degree two, generally expressed in the form:

f(x) = ax² + bx + c

where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve. The parabola opens upwards if a is positive and downwards if a is negative. The vertex of the parabola, the point where the curve changes direction, represents either the minimum or maximum value of the function. When a is negative, the parabola opens downwards, and the vertex represents the maximum value, which is the scenario we will focus on in this article.

The Given Function: f(x) = a + bx - cx²

Our specific quadratic function is given as:

f(x) = a + bx - cx²

where the constants a, b, and c are provided as:

  • a = 12
  • b = 4
  • c = 3

Substituting these values into the function, we get:

f(x) = 12 + 4x - 3x²

Now, our mission is to find the maximum value of this function. Since the coefficient of the x² term (-3) is negative, we know that the parabola opens downwards, and thus, the function has a maximum value at its vertex.

Method 1: Completing the Square

One effective method to determine the maximum value is by completing the square. This technique involves rewriting the quadratic function in vertex form, which directly reveals the vertex coordinates. The vertex form of a quadratic function is:

f(x) = a(x - h)² + k

where (h, k) represents the vertex of the parabola. The y-coordinate, k, gives us the maximum or minimum value of the function. Let's apply this method to our function:

f(x) = 12 + 4x - 3x²

  1. Rearrange the terms:

    First, rearrange the terms to group the x² and x terms together:

    f(x) = -3x² + 4x + 12

  2. Factor out the coefficient of x²:

    Factor out -3 from the first two terms:

    f(x) = -3(x² - (4/3)x) + 12

  3. Complete the square:

    To complete the square, we need to add and subtract the square of half the coefficient of the x term inside the parenthesis. The coefficient of x is -4/3, so half of it is -2/3, and the square of -2/3 is (4/9). Add and subtract this value inside the parenthesis:

    f(x) = -3(x² - (4/3)x + (4/9) - (4/9)) + 12

  4. Rewrite as a perfect square:

    Rewrite the expression inside the parenthesis as a perfect square:

    f(x) = -3((x - (2/3))² - (4/9)) + 12

  5. Distribute and simplify:

    Distribute the -3 and simplify the expression:

    f(x) = -3(x - (2/3))² + (4/3) + 12

    f(x) = -3(x - (2/3))² + (4/3) + (36/3)

    f(x) = -3(x - (2/3))² + (40/3)

Now, the function is in vertex form: f(x) = -3(x - (2/3))² + (40/3). The vertex of the parabola is (2/3, 40/3), and the maximum value of the function is 40/3.

To express this as a decimal rounded to two places:

40/3 ≈ 13.33

Method 2: Using the Vertex Formula

Another efficient method to find the vertex of a parabola is by using the vertex formula. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex (h) is given by:

h = -b / 2a

and the y-coordinate of the vertex (k), which is the maximum or minimum value, is found by substituting h back into the function:

k = f(h)

Let's apply this method to our function:

f(x) = -3x² + 4x + 12

  1. Identify a, b, and c:

    In our function, a = -3, b = 4, and c = 12.

  2. Calculate the x-coordinate of the vertex (h):

    h = -b / 2a = -4 / (2 * -3) = -4 / -6 = 2/3

  3. Calculate the y-coordinate of the vertex (k):

    Substitute h = 2/3 back into the function to find the maximum value:

    k = f(2/3) = -3(2/3)² + 4(2/3) + 12

    k = -3(4/9) + (8/3) + 12

    k = -(4/3) + (8/3) + 12

    k = (4/3) + 12

    k = (4/3) + (36/3)

    k = 40/3

Thus, the maximum value of the function is 40/3, which is approximately 13.33 when rounded to two decimal places.

Summary of Steps

To recap, here's a concise summary of the steps to find the maximum value of the quadratic function f(x) = 12 + 4x - 3x²:

  1. Write down the function: f(x) = 12 + 4x - 3x².

  2. Rearrange the function: f(x) = -3x² + 4x + 12.

  3. Choose a method:

    • Method 1: Completing the square
      • Factor out the coefficient of x².
      • Complete the square.
      • Rewrite in vertex form: f(x) = a(x - h)² + k.
      • The maximum value is k.
    • Method 2: Using the vertex formula
      • Identify a, b, and c.
      • Calculate h = -b / 2a.
      • Calculate k = f(h).
      • The maximum value is k.

  4. Calculate the maximum value.

  5. Round the final answer to two decimal places.

Conclusion

In this article, we've explored how to find the maximum value of a quadratic function. We examined two distinct methods: completing the square and using the vertex formula. Both approaches led us to the same result: the maximum value of the function f(x) = 12 + 4x - 3x² is 40/3, which is approximately 13.33 when rounded to two decimal places. Understanding these methods empowers you to solve a wide range of problems involving quadratic functions, making it a valuable skill in mathematics and related fields. Remember to practice these techniques to solidify your understanding and build confidence in your problem-solving abilities. Quadratic functions, with their unique properties and applications, are an integral part of the mathematical landscape.