Finding X-Intercepts Of Quadratic Function F(x) = X^2 + 6x + 5
How to find the x-intercepts of the function f(x) = x^2 + 6x + 5?
In the realm of mathematics, understanding the behavior of functions is crucial, especially when it comes to quadratic functions. These functions, represented by the general form f(x) = ax^2 + bx + c, where a, b, and c are constants, create a parabolic curve when graphed. A key aspect of analyzing these curves is identifying the x-intercepts, the points where the parabola intersects the x-axis. These points provide valuable information about the function's roots and overall shape. Let's delve into a specific example to illustrate the process of finding x-intercepts.
Understanding X-Intercepts
X-intercepts, also known as roots or zeros of a function, are the points where the graph of the function crosses the x-axis. At these points, the y-value or f(x) is equal to zero. Finding the x-intercepts of a quadratic function is equivalent to solving the quadratic equation ax^2 + bx + c = 0. There are several methods to accomplish this, including factoring, completing the square, and using the quadratic formula. Each method offers a unique approach, and the most suitable one often depends on the specific form of the equation.
For a quadratic function, the x-intercepts provide critical information about its behavior. They indicate where the function's value changes from positive to negative or vice versa. Moreover, the x-intercepts, along with the vertex of the parabola, help define the shape and position of the curve on the coordinate plane. In practical applications, x-intercepts can represent real-world scenarios such as the points where a projectile hits the ground or the break-even points in a business model.
Brianna's Graphing Task: f(x) = x^2 + 6x + 5
Consider Brianna's task: she needs to graph the function f(x) = x^2 + 6x + 5. To accurately graph this quadratic function, determining the x-intercepts is an essential step. The x-intercepts will provide Brianna with two key points on the parabola, allowing her to sketch the curve more precisely. By understanding where the function crosses the x-axis, Brianna can better visualize its behavior and characteristics. Let's explore how to find these crucial x-intercepts.
Finding the X-Intercepts by Factoring
The given function is f(x) = x^2 + 6x + 5. To find the x-intercepts, we need to solve the equation x^2 + 6x + 5 = 0. Factoring is a common and often efficient method for solving quadratic equations, especially when the equation can be easily factored. The goal of factoring is to rewrite the quadratic expression as a product of two binomials.
To factor the quadratic x^2 + 6x + 5, we look for two numbers that multiply to the constant term (5) and add up to the coefficient of the linear term (6). In this case, the numbers are 5 and 1 because 5 * 1 = 5 and 5 + 1 = 6. Thus, we can rewrite the quadratic expression as (x + 5)(x + 1). Now, the equation becomes (x + 5)(x + 1) = 0. To find the solutions for x, we set each factor equal to zero:
x + 5 = 0 or x + 1 = 0
Solving these linear equations gives us:
x = -5 and x = -1
These are the x-intercepts of the function f(x) = x^2 + 6x + 5. This means the parabola intersects the x-axis at the points (-5, 0) and (-1, 0). Brianna can now use these points as key references for graphing the function.
Alternative Methods for Finding X-Intercepts
While factoring is a straightforward method in this case, it's not always feasible for every quadratic equation. Other methods, such as the quadratic formula and completing the square, offer alternative approaches for finding x-intercepts. Understanding these methods provides a comprehensive toolkit for solving quadratic equations.
Using the Quadratic Formula
The quadratic formula is a universal method for finding the roots of any quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
For the function f(x) = x^2 + 6x + 5, we have a = 1, b = 6, and c = 5. Plugging these values into the quadratic formula, we get:
x = (-6 ± √(6^2 - 4 * 1 * 5)) / (2 * 1)
x = (-6 ± √(36 - 20)) / 2
x = (-6 ± √16) / 2
x = (-6 ± 4) / 2
This gives us two solutions:
x = (-6 + 4) / 2 = -1
x = (-6 - 4) / 2 = -5
These solutions confirm the x-intercepts are -1 and -5, consistent with the factoring method.
Completing the Square
Completing the square is another method for solving quadratic equations. It involves manipulating the equation to form a perfect square trinomial. For the function f(x) = x^2 + 6x + 5, we start by rewriting the equation as:
x^2 + 6x = -5
To complete the square, we take half of the coefficient of the x term (which is 6), square it (which is (6/2)^2 = 9), and add it to both sides of the equation:
x^2 + 6x + 9 = -5 + 9
(x + 3)^2 = 4
Taking the square root of both sides, we get:
x + 3 = ±2
Solving for x, we have:
x = -3 + 2 = -1
x = -3 - 2 = -5
Again, the x-intercepts are -1 and -5, reinforcing the accuracy of our results.
Determining the Correct X-Intercepts for Brianna's Graph
Having solved for the x-intercepts using multiple methods, we've consistently found that the x-intercepts for f(x) = x^2 + 6x + 5 are -5 and -1. This means that the correct answer Brianna should use for graphing the function is A. -5 and -1. These points are crucial for accurately plotting the parabola.
Understanding how to find x-intercepts is a fundamental skill in graphing quadratic functions. Whether you use factoring, the quadratic formula, or completing the square, the ability to determine these key points is essential for visualizing and analyzing the behavior of quadratic equations. In Brianna's case, knowing the x-intercepts helps her create an accurate graph of the function, allowing her to explore its properties and applications further.
Conclusion
In summary, finding the x-intercepts of a quadratic function is a critical step in understanding and graphing its parabolic curve. For the function f(x) = x^2 + 6x + 5, we've demonstrated how to find these intercepts using factoring, the quadratic formula, and completing the square. All methods lead to the same result: the x-intercepts are -5 and -1. These points not only define where the parabola crosses the x-axis but also provide a foundation for further analysis of the function's behavior and its applications in various mathematical and real-world contexts. Mastering these techniques empowers individuals like Brianna to confidently graph and interpret quadratic functions, unlocking a deeper understanding of mathematical concepts.