Vertex Form & Min Value: F(x)=2x²+8x+3

by Pedro Alvarez 39 views

Hey guys! Let's dive into the fascinating world of quadratic functions and explore how to determine the vertex form and find the maximum or minimum value of a given function. Today, we'll be dissecting the quadratic function f(x) = 2x² + 8x + 3. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Quadratic Function

Before we jump into the specifics, let's take a moment to appreciate the beauty and power 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 ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the leading coefficient a. If a > 0, the parabola opens upwards, and the function has a minimum value. Conversely, if a < 0, the parabola opens downwards, and the function has a maximum value.

In our case, the function f(x) = 2x² + 8x + 3 is a quadratic function with a = 2, b = 8, and c = 3. Since a = 2 is positive, we know that the parabola opens upwards, and the function has a minimum value. Our mission now is to find the vertex form of this function, which will reveal the coordinates of the vertex and the minimum value.

Transforming to Vertex Form: Completing the Square

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. The vertex form is incredibly useful because it directly tells us the vertex of the parabola, which is the point where the function attains its minimum (or maximum) value. To transform our given function f(x) = 2x² + 8x + 3 into vertex form, we'll employ a technique called completing the square.

Completing the square involves manipulating the quadratic expression to create a perfect square trinomial. Let's break down the steps:

  1. Factor out the leading coefficient from the first two terms:

    First, we factor out the coefficient of the x² term (which is 2 in our case) from the first two terms of the function:

    f(x) = 2(x² + 4x) + 3

  2. Complete the square inside the parentheses:

    To complete the square, we take half of the coefficient of the x term (which is 4), square it (4/2 = 2, 2² = 4), and add and subtract it inside the parentheses. This maintains the equality of the expression:

    f(x) = 2(x² + 4x + 4 - 4) + 3

  3. Rewrite the perfect square trinomial:

    The expression inside the parentheses, x² + 4x + 4, is a perfect square trinomial, which can be rewritten as (x + 2)². So, we have:

    f(x) = 2((x + 2)² - 4) + 3

  4. Distribute and simplify:

    Now, we distribute the 2 and simplify the expression:

    f(x) = 2(x + 2)² - 8 + 3

    f(x) = 2(x + 2)² - 5

Voilà! We have successfully transformed the function into vertex form: f(x) = 2(x + 2)² - 5.

Identifying the Vertex and Minimum Value

Now that we have the vertex form, it's a piece of cake to identify the vertex and the minimum value of the function. Comparing our result, f(x) = 2(x + 2)² - 5, with the general vertex form f(x) = a(x - h)² + k, we can see that:

  • h = -2
  • k = -5

Therefore, the vertex of the parabola is located at the point (-2, -5). Since the parabola opens upwards (because a = 2 is positive), the vertex represents the minimum point of the function. This means that the minimum value of f(x) is -5, which occurs when x = -2.

Summarizing the Findings

Let's recap what we've discovered:

  • The vertex form of the function f(x) = 2x² + 8x + 3 is f(x) = 2(x + 2)² - 5.
  • The vertex of the parabola is located at (-2, -5).
  • The function has a minimum value of -5.

Why is this Important?

Understanding the vertex form and finding the maximum or minimum value of a quadratic function has numerous applications in real-world scenarios. For instance:

  • Optimization problems: Quadratic functions are used to model various optimization problems, such as finding the maximum profit, minimizing costs, or determining the optimal trajectory of a projectile.
  • Engineering and physics: The parabolic shape of quadratic functions is fundamental in engineering and physics, appearing in the design of bridges, antennas, and projectile motion calculations.
  • Economics and finance: Quadratic functions can be used to model cost curves, revenue curves, and profit functions in economic and financial analysis.

Conclusion

Today, we've conquered the challenge of finding the vertex form and minimum value of the quadratic function f(x) = 2x² + 8x + 3. By mastering the technique of completing the square, we successfully transformed the function into vertex form, revealing the vertex coordinates and the minimum value. Remember, guys, the power of quadratic functions extends far beyond the classroom, making this a valuable skill to have in your mathematical toolkit. Keep exploring, keep learning, and keep conquering those mathematical mountains!

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