Finding X-Intercepts Of Quadratic Function F(x) = (1/2)x^2 + X - 9
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The realm of quadratic functions often presents intriguing challenges, especially when it comes to pinpointing the exact locations where these functions intersect the x-axis. In this comprehensive guide, we'll dissect the process of determining between which two ordered pairs the graph of a given quadratic function, specifically f(x) = (1/2)x² + x - 9, crosses the negative x-axis. We'll delve into the application of the quadratic formula, explore the significance of x-intercepts, and provide a step-by-step approach to solving this type of problem. This guide is designed to be your go-to resource, whether you're a student grappling with quadratic equations or an enthusiast eager to deepen your understanding of mathematical concepts.
Understanding the Significance of X-Intercepts
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To begin our journey, it's crucial to grasp the fundamental concept of x-intercepts. An x-intercept is the point where a graph intersects the x-axis. At these points, the y-coordinate is always zero. Therefore, finding the x-intercepts of a function involves solving for the values of x when f(x) = 0. These points hold immense significance in various mathematical and real-world applications. For instance, in physics, they might represent the points where a projectile hits the ground, while in economics, they could indicate the break-even points for a business. The x-intercepts provide crucial insights into the behavior and characteristics of a function.
In the context of quadratic functions, the x-intercepts are also known as the roots or zeros of the function. A quadratic function, defined by the general form f(x) = ax² + bx + c, can have up to two x-intercepts, one x-intercept, or no x-intercepts, depending on the discriminant (b² - 4ac). This discriminant plays a pivotal role in determining the nature of the roots and the overall shape of the parabola. Understanding the x-intercepts allows us to visualize the graph of the quadratic function and predict its behavior.
To find the x-intercepts, we set f(x) to zero and solve the resulting quadratic equation. This often involves using the quadratic formula, factoring, or completing the square. Each method offers a unique approach to unraveling the roots of the equation. The x-intercepts not only provide specific points on the graph but also serve as key landmarks for understanding the function's symmetry and concavity. By analyzing the x-intercepts, we can gain a deeper understanding of the quadratic function's characteristics and its relevance in various contexts.
The Quadratic Formula: A Powerful Tool
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The quadratic formula is an indispensable tool for solving quadratic equations of the form ax² + bx + c = 0. This formula provides a direct method for finding the roots, regardless of whether the equation can be easily factored. The quadratic formula is expressed as: x = (-b ± √(b² - 4ac)) / 2a. Here, a, b, and c are the coefficients of the quadratic equation, and the symbol '±' indicates that there are potentially two solutions.
This formula is derived by completing the square on the general quadratic equation, and it guarantees a solution, even when the roots are irrational or complex. The part of the formula under the square root, b² - 4ac, is known as the discriminant. The discriminant determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. Understanding the discriminant is crucial for interpreting the solutions obtained from the quadratic formula.
The quadratic formula is not just a mathematical tool; it's a gateway to understanding the behavior of quadratic functions. By plugging in the coefficients of a specific quadratic equation, we can quickly determine its roots, which correspond to the x-intercepts of its graph. These x-intercepts provide critical information about the parabola's position and orientation in the coordinate plane. Whether the parabola opens upwards or downwards, and whether it intersects the x-axis at two points, one point, or not at all, can be deduced from the solutions obtained using the quadratic formula.
Applying the Quadratic Formula to Our Function
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Now, let's apply the quadratic formula to the given function, f(x) = (1/2)x² + x - 9. To find the x-intercepts, we need to solve the equation (1/2)x² + x - 9 = 0. Identifying the coefficients, we have a = 1/2, b = 1, and c = -9. Plugging these values into the quadratic formula, we get:
x = (-1 ± √(1² - 4 * (1/2) * (-9))) / (2 * (1/2)).
Simplifying the expression under the square root, we have:
x = (-1 ± √(1 + 18)) / 1.
This further simplifies to:
x = (-1 ± √19) / 1.
Therefore, the two solutions are:
x₁ = -1 + √19 and x₂ = -1 - √19.
These values represent the x-coordinates of the points where the graph of the function intersects the x-axis. To determine between which ordered pairs the graph crosses the negative x-axis, we need to approximate these values. The square root of 19 is approximately 4.36. Therefore, x₁ ≈ -1 + 4.36 ≈ 3.36 and x₂ ≈ -1 - 4.36 ≈ -5.36. This gives us a positive x-intercept around 3.36 and a negative x-intercept around -5.36. The negative x-intercept is the one we're interested in for this problem.
By carefully applying the quadratic formula, we've successfully identified the x-intercepts of the given function. The next step is to analyze these values and determine between which ordered pairs the graph crosses the negative x-axis. This involves understanding the numerical values of the roots and their implications for the graph's behavior. The quadratic formula has provided us with the precise locations where the function intersects the x-axis, enabling us to address the original question effectively.
Determining the Interval for the Negative X-Intercept
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Having calculated the two x-intercepts, x₁ ≈ 3.36 and x₂ ≈ -5.36, we now focus on the negative x-intercept, x₂ ≈ -5.36. The question asks between which two ordered pairs the graph crosses the negative x-axis. Since we're looking at x-intercepts, the y-coordinate will be 0. Thus, we are looking for two ordered pairs of the form (x, 0).
The negative x-intercept of approximately -5.36 lies between the integers -6 and -5. Therefore, the graph of the function crosses the negative x-axis between the ordered pairs (-6, 0) and (-5, 0). This means that the point where the graph intersects the x-axis on the negative side is located somewhere between these two points on the x-axis. This understanding is crucial for visualizing the graph of the quadratic function and its behavior in the coordinate plane.
The ordered pairs (-6, 0) and (-5, 0) define an interval on the x-axis within which the graph of the function transitions from negative y-values to positive y-values, or vice versa. This transition occurs precisely at the x-intercept. By identifying this interval, we gain a clearer picture of the function's behavior around its roots. This is a fundamental aspect of analyzing quadratic functions and their graphs. The ability to pinpoint the interval containing the x-intercept demonstrates a solid understanding of the relationship between the algebraic solutions and the graphical representation of the function.
In summary, by calculating the roots using the quadratic formula and then analyzing the numerical value of the negative root, we've successfully determined that the graph of f(x) = (1/2)x² + x - 9 crosses the negative x-axis between the ordered pairs (-6, 0) and (-5, 0). This process exemplifies the power of combining algebraic techniques with graphical reasoning to solve mathematical problems.
Conclusion: Mastering Quadratic Functions
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In this comprehensive guide, we've meticulously explored the process of determining between which two ordered pairs the graph of the quadratic function f(x) = (1/2)x² + x - 9 crosses the negative x-axis. We began by emphasizing the significance of x-intercepts and their role in understanding the behavior of functions. We then introduced the quadratic formula, a powerful tool for solving quadratic equations, and demonstrated its application to our specific function. Through careful calculation and analysis, we found the x-intercepts and pinpointed the interval containing the negative x-intercept.
This journey through quadratic functions highlights the importance of understanding fundamental concepts and mastering key techniques. The quadratic formula is not just a formula to memorize; it's a gateway to understanding the roots of quadratic equations and their graphical representations. By applying the quadratic formula and interpreting the results, we gain valuable insights into the function's behavior and its relationship to the coordinate plane. The ability to determine the interval containing the x-intercepts demonstrates a solid grasp of both algebraic and graphical concepts.
Mastering quadratic functions is not only essential for success in mathematics but also provides a foundation for tackling more advanced topics. The concepts and techniques discussed in this guide are applicable to a wide range of problems in various fields, from physics and engineering to economics and computer science. By developing a deep understanding of quadratic functions, you'll be well-equipped to tackle future mathematical challenges and apply your knowledge to real-world scenarios. The ability to confidently solve quadratic equations and interpret their solutions is a valuable skill that will serve you well in your academic and professional pursuits.
