Finding The Vertex And X-Intercepts Of A Quadratic Function F(x) = X² - 4x - 5

In this article, we will delve into the process of finding the vertex and x-intercepts of a quadratic function. Specifically, we will focus on the function f(x) = x² - 4x - 5. Understanding these key features allows us to effectively graph and analyze quadratic equations. Let's embark on this mathematical journey together!

Determining the Vertex of the Quadratic Function

The vertex of a parabola, which is the graphical representation of a quadratic function, is a crucial point. It represents either the minimum or maximum value of the function. For a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. In our case, the function is f(x) = x² - 4x - 5, where a = 1, b = -4, and c = -5.

Applying the formula, we get x = -(-4) / (2 * 1) = 4 / 2 = 2. This means the x-coordinate of the vertex is 2. To find the corresponding y-coordinate, we substitute x = 2 back into the original function: f(2) = (2)² - 4(2) - 5 = 4 - 8 - 5 = -9. Therefore, the vertex of the parabola is at the point (2, -9). This point is significant as it signifies the minimum value of the function since the coefficient of the x² term (a = 1) is positive, indicating the parabola opens upwards.

Understanding how to calculate the vertex is pivotal in analyzing quadratic functions. The vertex not only gives us the minimum or maximum value but also the axis of symmetry, which is a vertical line passing through the vertex. This axis divides the parabola into two symmetrical halves, making it easier to sketch the graph and understand the function's behavior. Moreover, the vertex form of a quadratic equation, f(x) = a(x - h)² + k, directly reveals the vertex coordinates as (h, k), providing an alternative method for identifying the vertex. For our function, converting to vertex form can give us a different perspective on finding the same critical point.

Plotting the Vertex on the Coordinate Plane

Once we've determined the vertex coordinates, the next step is to plot this point on the coordinate plane. In our case, the vertex is located at (2, -9). This means we move 2 units to the right along the x-axis and 9 units down along the y-axis from the origin (0, 0). Marking this point on the graph provides a crucial reference for sketching the parabola. The vertex serves as the turning point of the parabola, and its position helps define the overall shape and orientation of the curve.

Plotting the vertex accurately is essential for visualizing the quadratic function. It helps us understand where the parabola reaches its minimum or maximum value and how it opens in relation to the x-axis. In addition to the vertex, plotting other points, such as the y-intercept and x-intercepts (which we will find next), gives us a more detailed picture of the parabola's graph. When graphing by hand, starting with the vertex is often the best approach as it gives you a central point around which to build the rest of the curve. If using graphing software or a calculator, inputting the function f(x) = x² - 4x - 5 will display the parabola, clearly showing the vertex at (2, -9).

The process of plotting the vertex is not just a mechanical step; it's a visual aid that enhances our comprehension of the function's behavior. By seeing the vertex on the graph, we can immediately infer whether the parabola opens upwards or downwards and where its extreme value lies. This visual representation is invaluable for problem-solving and for understanding the real-world applications of quadratic functions, such as in projectile motion or optimization problems. The act of plotting reinforces the connection between the algebraic representation of the function and its geometric interpretation.

Factoring to Find the X-Intercepts

The x-intercepts, also known as the roots or zeros of the function, are the points where the parabola intersects the x-axis. At these points, the value of the function f(x) is equal to zero. To find the x-intercepts of f(x) = x² - 4x - 5, we need to solve the equation x² - 4x - 5 = 0. Factoring is a common method for solving quadratic equations, and it involves expressing the quadratic expression as a product of two binomials.

In this case, we are looking for two numbers that multiply to -5 (the constant term) and add up to -4 (the coefficient of the x term). The numbers -5 and 1 satisfy these conditions, since (-5) * 1 = -5 and (-5) + 1 = -4. Therefore, we can factor the quadratic expression as (x - 5)(x + 1) = 0. To find the x-intercepts, we set each factor equal to zero and solve for x:

• x - 5 = 0 => x = 5
• x + 1 = 0 => x = -1

Thus, the x-intercepts are x = 5 and x = -1. These values represent the points where the parabola crosses the x-axis. In coordinate form, these points are (5, 0) and (-1, 0). Understanding how to find the x-intercepts is crucial for sketching the parabola and for solving various mathematical and real-world problems. The x-intercepts give us valuable information about the function's behavior and its solutions.

Factoring is a powerful tool, but it's not always the easiest or most applicable method for finding x-intercepts. When a quadratic equation is not easily factorable, other techniques such as using the quadratic formula or completing the square can be employed. The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, provides a universal solution for any quadratic equation in the form ax² + bx + c = 0. The discriminant, b² - 4ac, within the formula, tells us the nature of the roots: if it's positive, there are two real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are no real roots (the roots are complex). For our equation, the discriminant is (-4)² - 4(1)(-5) = 16 + 20 = 36, which is positive, confirming our two real x-intercepts.

Identifying the X-Intercepts

As we found through factoring, the x-intercepts of the function f(x) = x² - 4x - 5 are x = 5 and x = -1. These points, (5, 0) and (-1, 0), are where the parabola intersects the x-axis. They are also the solutions or roots of the quadratic equation x² - 4x - 5 = 0. Understanding these intercepts is vital for sketching the graph of the parabola and for interpreting the function's behavior.

Knowing the x-intercepts, along with the vertex, gives us a clear picture of the parabola's shape and position on the coordinate plane. The x-intercepts tell us where the function's value is zero, which can have significant meaning in real-world applications. For example, if this function represented the trajectory of a projectile, the x-intercepts would represent the points where the projectile lands on the ground. Moreover, the x-intercepts can be used to determine the intervals where the function is positive or negative, providing further insight into its characteristics.

In summary, finding the vertex and x-intercepts of a quadratic function are fundamental steps in analyzing and graphing parabolas. The vertex represents the extreme point, and the x-intercepts indicate where the function crosses the x-axis. By combining these elements, we can effectively sketch the graph, understand the function's behavior, and solve related problems. Whether through factoring, the quadratic formula, or other methods, mastering these techniques is essential for a comprehensive understanding of quadratic functions.