Finding X-Intercepts Of The Quadratic Function Y=x^2-4x-5

In the realm of mathematics, quadratic functions hold a significant position. They are defined as polynomial functions of the second degree, 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 that opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the features of a parabola is crucial in various applications, including physics, engineering, and economics. One of the key features of a parabola is its x-intercepts, also known as roots or zeros. These are the points where the parabola intersects the x-axis, representing the values of x for which the function f(x) equals zero. Finding the x-intercepts is a fundamental task in analyzing quadratic functions, as it provides valuable information about the function's behavior and its relationship to the x-axis. There are several methods to determine the x-intercepts of a quadratic function, including factoring, using the quadratic formula, and completing the square. Each method offers a unique approach, and the choice of method often depends on the specific form of the quadratic equation and the individual's preference. In this article, we will delve into the process of finding the x-intercepts of a given quadratic function, y = x² - 4x - 5, and explore the different techniques that can be employed. By mastering the concept of x-intercepts and the methods to find them, you will gain a deeper understanding of quadratic functions and their applications in various fields.

Determining the x-intercepts by Factoring

Factoring is a powerful technique for finding the x-intercepts of a quadratic function, particularly when the equation can be easily factored. The core idea behind factoring is to rewrite the quadratic expression as a product of two linear expressions. This transformation allows us to identify the values of x that make the expression equal to zero, which correspond to the x-intercepts. In the given quadratic function, y = x² - 4x - 5, we aim to factor the expression x² - 4x - 5. To do this, we need to find two numbers that multiply to -5 (the constant term) and add up to -4 (the coefficient of the x term). By careful consideration, we can identify these two numbers as -5 and 1. Indeed, -5 multiplied by 1 equals -5, and -5 plus 1 equals -4. Now, we can rewrite the quadratic expression using these two numbers: x² - 4x - 5 = (x - 5)(x + 1). This factored form reveals the two linear expressions, (x - 5) and (x + 1), whose product equals the original quadratic expression. To find the x-intercepts, we set the factored expression equal to zero: (x - 5)(x + 1) = 0. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x: x - 5 = 0 or x + 1 = 0. Solving the first equation, x - 5 = 0, we add 5 to both sides, resulting in x = 5. Solving the second equation, x + 1 = 0, we subtract 1 from both sides, resulting in x = -1. Thus, we have found the two x-intercepts of the quadratic function: x = 5 and x = -1. These values represent the points where the parabola intersects the x-axis. Factoring is an efficient method for finding x-intercepts when the quadratic expression can be readily factored. However, not all quadratic expressions are factorable using simple integer coefficients. In such cases, alternative methods like the quadratic formula or completing the square become necessary.

Matching the Function with its Graph

After determining the x-intercepts, the next crucial step is to match the quadratic function with its graph. This involves understanding how the x-intercepts, along with other key features of the function, shape the parabola and its position on the coordinate plane. The x-intercepts, as we have already established, are the points where the parabola intersects the x-axis. These points provide valuable information about the function's roots or zeros. In our example, the x-intercepts are x = 5 and x = -1. This means the parabola will pass through the points (5, 0) and (-1, 0) on the coordinate plane. Another critical feature of a parabola is its vertex, which represents the highest or lowest point on the curve. The vertex is the turning point of the parabola, and its coordinates can be determined using the formula x = -b / 2a, where 'a' and 'b' are the coefficients of the quadratic function in the standard form f(x) = ax² + bx + c. In our case, a = 1 and b = -4, so the x-coordinate of the vertex is x = -(-4) / (2 * 1) = 2. To find the y-coordinate of the vertex, we substitute x = 2 back into the original function: y = (2)² - 4(2) - 5 = 4 - 8 - 5 = -9. Therefore, the vertex of the parabola is at the point (2, -9). The y-intercept is another important feature, representing the point where the parabola intersects the y-axis. To find the y-intercept, we set x = 0 in the quadratic function: y = (0)² - 4(0) - 5 = -5. So, the y-intercept is at the point (0, -5). Now, with the x-intercepts (5, 0) and (-1, 0), the vertex (2, -9), and the y-intercept (0, -5), we have enough information to sketch the graph of the parabola. Since the coefficient of the x² term (a) is positive (a = 1), the parabola opens upwards. This means the vertex is the minimum point of the parabola. By plotting these key points and considering the direction of the parabola, we can accurately match the function y = x² - 4x - 5 with its corresponding graph. The graph will be a U-shaped curve passing through the x-intercepts at x = 5 and x = -1, with its vertex at (2, -9) and y-intercept at (0, -5). Matching the function with its graph is a visual way to confirm our calculations and gain a deeper understanding of the relationship between the algebraic representation and the geometric representation of the quadratic function.

Alternative Methods for Finding x-intercepts

While factoring is a convenient method for finding x-intercepts when applicable, other methods can be employed when factoring proves challenging or impossible. Two prominent alternatives are the quadratic formula and completing the square. The quadratic formula is a universal solution for finding the roots of any quadratic equation, regardless of its factorability. It is derived from the process of completing the square and provides a direct formula for calculating the x-intercepts. The quadratic formula is expressed as: x = (-b ± √(b² - 4ac)) / 2a, where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. In our example, y = x² - 4x - 5, we have a = 1, b = -4, and c = -5. Substituting these values into the quadratic formula, we get: x = (-(-4) ± √((-4)² - 4 * 1 * -5)) / (2 * 1). Simplifying this expression, we have: x = (4 ± √(16 + 20)) / 2 = (4 ± √36) / 2 = (4 ± 6) / 2. This gives us two possible solutions for x: x = (4 + 6) / 2 = 10 / 2 = 5 and x = (4 - 6) / 2 = -2 / 2 = -1. These are the same x-intercepts we found earlier using factoring, confirming the consistency of the quadratic formula. Completing the square is another method for finding x-intercepts, which involves transforming the quadratic equation into a perfect square trinomial. This technique is particularly useful when the quadratic expression is not easily factorable. To complete the square for the equation y = x² - 4x - 5, we first move the constant term to the right side: x² - 4x = 5. Next, we take half of the coefficient of the x term (-4), square it ((-2)² = 4), and add it to both sides of the equation: x² - 4x + 4 = 5 + 4. Now, the left side is a perfect square trinomial, which can be factored as: (x - 2)² = 9. Taking the square root of both sides, we get: x - 2 = ±√9 = ±3. This gives us two possible equations: x - 2 = 3 or x - 2 = -3. Solving these equations, we find: x = 5 and x = -1, which are the same x-intercepts we obtained using factoring and the quadratic formula. The quadratic formula and completing the square provide alternative approaches to finding x-intercepts when factoring is not straightforward. The quadratic formula is a direct and reliable method, while completing the square offers a deeper understanding of the structure of quadratic equations.

Conclusion

In conclusion, finding the x-intercepts of a quadratic function is a fundamental task in mathematics with various applications. We explored three methods for determining x-intercepts: factoring, using the quadratic formula, and completing the square. Factoring is an efficient method when the quadratic expression can be readily factored, while the quadratic formula provides a universal solution for any quadratic equation. Completing the square offers a deeper understanding of the structure of quadratic equations and can be useful in various contexts. In the specific example of the quadratic function y = x² - 4x - 5, we successfully found the x-intercepts using factoring, confirming the solutions with the quadratic formula. The x-intercepts were determined to be x = 5 and x = -1, representing the points where the parabola intersects the x-axis. Furthermore, we discussed the importance of matching the function with its graph, highlighting the role of x-intercepts, the vertex, and the y-intercept in shaping the parabola and its position on the coordinate plane. By understanding these key features, we can accurately visualize the graph of the quadratic function and its relationship to the x-axis. Mastering the techniques for finding x-intercepts and understanding the graphical representation of quadratic functions are essential skills for students and professionals in various fields, including mathematics, physics, engineering, and economics. The ability to analyze quadratic functions and their properties opens doors to solving real-world problems and making informed decisions. Whether it's modeling projectile motion, optimizing business processes, or understanding financial trends, quadratic functions play a significant role in our world. By continuing to explore and practice these concepts, you will enhance your mathematical skills and gain a deeper appreciation for the power and elegance of quadratic functions.