Vertex Form Of F(x) = 8x² + 4x Explained

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In the realm of mathematics, quadratic functions play a pivotal role, appearing in various applications ranging from physics to engineering. These functions, characterized by their parabolic curves, can be expressed in different forms, each offering unique insights into their behavior. Among these forms, the vertex form stands out as a powerful tool for understanding the key features of a quadratic function, such as its vertex (the point of maximum or minimum value) and axis of symmetry. This article delves into the process of converting the standard form of a quadratic function, specifically f(x) = 8x² + 4x, into its vertex form, and explores the significance of this transformation.

What is the Vertex Form of a Quadratic Function?

Before we embark on the conversion process, it's crucial to understand what the vertex form actually represents. The vertex form of a quadratic function is expressed as:

f(x) = a(x - h)² + k

Where:

  • a determines the direction and steepness of the parabola.
  • (h, k) represents the coordinates of the vertex.

The vertex form provides a direct way to identify the vertex of the parabola, which is the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value. The h value represents the x-coordinate of the vertex, and the k value represents the y-coordinate, which is the minimum or maximum value of the function.

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is given by x = h.

Converting f(x) = 8x² + 4x to Vertex Form: Completing the Square

The standard form of a quadratic function is given by:

f(x) = ax² + bx + c

Our given function, f(x) = 8x² + 4x, is in this standard form, where a = 8, b = 4, and c = 0. To convert this to vertex form, we employ a technique called completing the square.

Completing the square involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a squared binomial. Here's a step-by-step breakdown of the process:

  1. Factor out the coefficient of x² from the first two terms:

    f(x) = 8(x² + (1/2)x)

    This step ensures that the coefficient of the x² term inside the parentheses is 1, which is necessary for completing the square.

  2. Complete the square inside the parentheses:

    To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term inside the parentheses is 1/2, so half of it is 1/4, and the square of 1/4 is 1/16.

    f(x) = 8(x² + (1/2)x + 1/16 - 1/16)

    We've added and subtracted 1/16 inside the parentheses, which doesn't change the overall value of the expression.

  3. Rewrite the perfect square trinomial as a squared binomial:

    The first three terms inside the parentheses now form a perfect square trinomial, which can be factored as follows:

    f(x) = 8((x + 1/4)² - 1/16)

  4. Distribute the coefficient outside the parentheses:

    f(x) = 8(x + 1/4)² - 8(1/16)

  5. Simplify:

    f(x) = 8(x + 1/4)² - 1/2

Now, the function is in vertex form: f(x) = 8(x + 1/4)² - 1/2.

Identifying the Vertex and Axis of Symmetry

Comparing our result with the general vertex form f(x) = a(x - h)² + k, we can identify the following:

  • a = 8
  • h = -1/4
  • k = -1/2

Therefore, the vertex of the parabola is at the point (-1/4, -1/2). Since a = 8 is positive, the parabola opens upwards, and the vertex represents the minimum point of the function.

The axis of symmetry is a vertical line passing through the vertex, and its equation is x = h. In this case, the axis of symmetry is x = -1/4.

Significance of the Vertex Form

The vertex form provides valuable insights into the behavior of the quadratic function. By simply looking at the vertex form, we can directly determine:

  • The vertex of the parabola (the point of minimum or maximum value).
  • The axis of symmetry.
  • Whether the parabola opens upwards or downwards (based on the sign of a).
  • The minimum or maximum value of the function (the k value).

These pieces of information are crucial for sketching the graph of the quadratic function and for solving various application problems.

For instance, in physics, quadratic functions are used to model projectile motion. The vertex of the parabola represents the highest point reached by the projectile, and the axis of symmetry indicates the time at which this maximum height is achieved. Similarly, in engineering, quadratic functions can be used to design parabolic reflectors, where the vertex represents the focal point of the reflector.

Alternative Method: Using Formulas for Vertex Coordinates

While completing the square is a fundamental technique, there's an alternative method to find the vertex coordinates directly using formulas. For a quadratic function in standard form f(x) = ax² + bx + c, the coordinates of the vertex (h, k) can be found using the following formulas:

  • h = -b / 2a
  • k = f(h)

Let's apply these formulas to our function f(x) = 8x² + 4x:

  1. Calculate h:

    h = -b / 2a = -4 / (2 * 8) = -4 / 16 = -1/4

  2. Calculate k:

    k = f(h) = f(-1/4) = 8(-1/4)² + 4(-1/4) = 8(1/16) - 1 = 1/2 - 1 = -1/2

We obtain the same vertex coordinates (-1/4, -1/2) as we did by completing the square. This method provides a quicker way to find the vertex, especially when dealing with more complex quadratic functions.

Graphing the Function

Knowing the vertex and the axis of symmetry, we can easily sketch the graph of f(x) = 8x² + 4x. The vertex (-1/4, -1/2) is the minimum point of the parabola, and the axis of symmetry is the vertical line x = -1/4. Since a = 8 is positive, the parabola opens upwards.

To get a more accurate graph, we can find a few additional points by plugging in different values of x into the function. For example:

  • f(0) = 8(0)² + 4(0) = 0
  • f(-1) = 8(-1)² + 4(-1) = 8 - 4 = 4

Plotting these points and using the symmetry of the parabola, we can draw a smooth curve that represents the graph of f(x) = 8x² + 4x.

Applications and Real-World Examples

Quadratic functions, and their vertex forms, have numerous applications in various fields. Here are a few examples:

  • Projectile Motion: As mentioned earlier, the path of a projectile (such as a ball thrown in the air) can be modeled using a quadratic function. The vertex represents the maximum height reached by the projectile.
  • Optimization Problems: Many optimization problems in calculus involve finding the maximum or minimum value of a function. Quadratic functions are often used in these problems, and the vertex represents the optimal solution.
  • Engineering Design: Parabolic shapes are used in various engineering designs, such as satellite dishes, bridges, and arches. The vertex of the parabola plays a crucial role in these designs.
  • Economics: Quadratic functions can be used to model cost, revenue, and profit functions in economics. The vertex can help determine the point of maximum profit or minimum cost.

Conclusion

Converting a quadratic function to vertex form is a valuable skill in mathematics. It allows us to easily identify the vertex, axis of symmetry, and the minimum or maximum value of the function. By understanding the significance of the vertex form, we can gain deeper insights into the behavior of quadratic functions and their applications in various fields. Whether using completing the square or the formulas for vertex coordinates, mastering this conversion empowers us to analyze and solve a wide range of mathematical problems.

In summary, the function f(x) = 8x² + 4x in vertex form is f(x) = 8(x + 1/4)² - 1/2, with a vertex at (-1/4, -1/2) and an axis of symmetry x = -1/4. This transformation provides a clear understanding of the function's key characteristics and its potential applications.