Find The Vertex Of F(x) = 5x² - 20x - 8 A Step-by-Step Guide

Finding the vertex of a quadratic function is a fundamental skill in algebra, with applications ranging from optimizing projectile motion to determining the minimum cost in business scenarios. In this guide, we'll break down the process step-by-step, using the example function f(x) = 5x² - 20x - 8. We'll explore different methods, ensuring you have a solid grasp of how to find the vertex, no matter the quadratic function you encounter. So, let's dive in and master this essential mathematical concept, guys!

Understanding Quadratic Functions and the Vertex

Before we jump into the calculations, let's take a moment to understand what a quadratic function is and what the vertex represents. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

Where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic function is a parabola, a U-shaped curve. This parabola can open upwards (if a > 0) or downwards (if a < 0).

The vertex is the point where the parabola changes direction. It's the minimum point if the parabola opens upwards and the maximum point if the parabola opens downwards. The vertex is a crucial feature of the parabola, providing valuable information about the function's behavior. Specifically, the vertex gives us the axis of symmetry, which is a vertical line passing through the vertex that divides the parabola into two symmetrical halves. The x-coordinate of the vertex also tells us the x-value where the function reaches its minimum or maximum value. Understanding these basic concepts is crucial to effectively find the vertex and apply this knowledge in various problem-solving situations.

In our example, f(x) = 5x² - 20x - 8, we have a = 5, b = -20, and c = -8. Since a is positive (5 > 0), the parabola opens upwards, and the vertex will be the minimum point.

Method 1: Using the Vertex Formula

The most direct method to find the vertex is by using the vertex formula. This formula provides the x-coordinate of the vertex directly, which we can then use to find the y-coordinate. The vertex formula is derived from completing the square (another method we'll discuss later), and it's a handy tool to have in your mathematical arsenal.

The x-coordinate of the vertex, often denoted as h, is given by:

h = -b / 2a

Where a and b are the coefficients from the quadratic function f(x) = ax² + bx + c. Once we have the x-coordinate (h), we can find the y-coordinate, often denoted as k, by substituting h back into the original function:

k = f(h)

Therefore, the vertex is the point (h, k).

Let's apply this formula to our example, f(x) = 5x² - 20x - 8. We have a = 5 and b = -20. Plugging these values into the vertex formula, we get:

h = -(-20) / (2 * 5) = 20 / 10 = 2

So, the x-coordinate of the vertex is 2. Now, we substitute this value back into the original function to find the y-coordinate:

k = f(2) = 5(2)² - 20(2) - 8 = 5(4) - 40 - 8 = 20 - 40 - 8 = -28

Thus, the y-coordinate of the vertex is -28. Therefore, the vertex of the quadratic function f(x) = 5x² - 20x - 8 is (2, -28). This method is highly efficient and generally the quickest way to find the vertex, especially when you're dealing with a standard quadratic equation.

Method 2: Completing the Square

Another powerful technique to find the vertex is by completing the square. This method not only helps us locate the vertex but also transforms the quadratic function into vertex form, which provides valuable insights into the function's properties. Completing the square involves rewriting the quadratic expression ax² + bx + c in the form a(x - h)² + k, where (h, k) is the vertex.

Here's how we can apply completing the square to our example, f(x) = 5x² - 20x - 8:

1. Factor out the coefficient of x² (if it's not 1) from the first two terms: f(x) = 5(x² - 4x) - 8

2. Complete the square inside the parentheses: To complete the square for x² - 4x, we take half of the coefficient of the x term (-4), square it ((-2)² = 4), and add and subtract it inside the parentheses: f(x) = 5(x² - 4x + 4 - 4) - 8

3. Rewrite the perfect square trinomial: The expression x² - 4x + 4 is a perfect square and can be written as (x - 2)²: f(x) = 5((x - 2)² - 4) - 8

4. Distribute and simplify: f(x) = 5(x - 2)² - 20 - 8 f(x) = 5(x - 2)² - 28

Now, the function is in vertex form, f(x) = a(x - h)² + k, where a = 5, h = 2, and k = -28. The vertex is therefore (2, -28), which matches the result we obtained using the vertex formula. Completing the square is a bit more involved than the vertex formula, but it's a valuable skill as it provides the vertex form directly, making it easier to analyze transformations and other properties of the quadratic function.

Method 3: Using Calculus (For the Curious Minds!)

For those familiar with calculus, there's yet another way to find the vertex. The vertex represents the point where the slope of the parabola is zero. In calculus, the slope of a function at a point is given by its derivative. Therefore, we can find the x-coordinate of the vertex by finding the derivative of the quadratic function and setting it equal to zero.

Let's apply this method to our example, f(x) = 5x² - 20x - 8:

1. Find the derivative of f(x): Using the power rule, the derivative f'(x) is: f'(x) = 10x - 20

2. Set the derivative equal to zero and solve for x: 10x - 20 = 0 10x = 20 x = 2

So, the x-coordinate of the vertex is 2. Now, we substitute this value back into the original function to find the y-coordinate:

f(2) = 5(2)² - 20(2) - 8 = -28

Again, we find that the vertex is (2, -28). While calculus provides an alternative approach, it's generally more efficient to use the vertex formula or completing the square for quadratic functions. However, this method demonstrates the connection between algebra and calculus and can be useful for understanding more complex functions.

Putting It All Together

We've explored three different methods to find the vertex of a quadratic function: the vertex formula, completing the square, and using calculus. Each method has its strengths and weaknesses, but they all lead to the same result. For our example function, f(x) = 5x² - 20x - 8, we consistently found the vertex to be (2, -28).

To recap:

• The vertex formula (h = -b / 2a) is the most direct and efficient method.
• Completing the square transforms the function into vertex form, which is useful for understanding transformations and other properties.
• Calculus provides an alternative approach using derivatives, but it's generally more complex for quadratic functions.

No matter which method you choose, mastering the skill of finding the vertex will empower you to analyze quadratic functions effectively and solve a wide range of problems in mathematics and beyond. So, keep practicing, and you'll become a vertex-finding pro in no time, guys! Remember, understanding the underlying concepts and practicing different methods is key to mastering any mathematical skill.

In this comprehensive guide, we have thoroughly explored various methods to find the vertex of a quadratic function. Whether you prefer the straightforward approach of the vertex formula, the insightful process of completing the square, or the calculus-based method, you now have a solid understanding of how to locate this crucial point on a parabola. The vertex is not just a point; it's a key to unlocking the behavior and properties of quadratic functions. By mastering this skill, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications. Keep practicing, and you'll confidently find the vertex of any quadratic function that comes your way!