Finding X-Intercepts Of Quadratic Functions An Illustrated Guide

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In the realm of mathematics, quadratic functions hold a prominent position, shaping the curves of parabolas and playing a crucial role in various real-world applications. Understanding the behavior of these functions, particularly their intersections with the x-axis, is essential for comprehending their properties and applications. In this comprehensive guide, we embark on a journey to explore the intricacies of finding the x-intercepts of a quadratic function, focusing on the specific function f(x) = (1/2)x² + x - 9. We'll delve into the methods, calculations, and interpretations involved, ensuring a clear and concise understanding of this fundamental concept.

Decoding the X-Intercepts: Where the Parabola Meets the Axis

The x-intercepts, also known as the roots or zeros of a function, are the points where the graph of the function intersects the x-axis. At these points, the y-value of the function is zero. Identifying the x-intercepts provides valuable insights into the function's behavior, including its symmetry, minimum or maximum value, and the regions where the function is positive or negative. For a quadratic function, the x-intercepts correspond to the solutions of the quadratic equation formed when the function is set equal to zero.

To find the x-intercepts of the given quadratic function, f(x) = (1/2)x² + x - 9, we need to solve the equation (1/2)x² + x - 9 = 0. This equation represents a parabola, and its x-intercepts indicate where the parabola crosses the x-axis. To determine the intervals between which the graph crosses the negative x-axis, we will first find the exact x-intercepts using the quadratic formula. This formula is a powerful tool that provides a general solution for quadratic equations of the form ax² + bx + c = 0.

The Quadratic Formula: A Key to Unlocking the Roots

The quadratic formula is a cornerstone of algebra, providing a universal method for solving quadratic equations. It states that for a quadratic equation of the form ax² + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

In this formula, 'a', 'b', and 'c' are the coefficients of the quadratic equation. The term under the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots:

  • If b² - 4ac > 0, the equation has two distinct real roots, indicating that the parabola intersects the x-axis at two distinct points.
  • If b² - 4ac = 0, the equation has one real root (a repeated root), indicating that the parabola touches the x-axis at one point.
  • If b² - 4ac < 0, the equation has no real roots, indicating that the parabola does not intersect the x-axis.

In our case, the quadratic equation is (1/2)x² + x - 9 = 0. To simplify calculations, we can multiply the entire equation by 2 to eliminate the fraction, resulting in x² + 2x - 18 = 0. Now, we can identify the coefficients as a = 1, b = 2, and c = -18. Plugging these values into the quadratic formula, we get:

x = (-2 ± √(2² - 4 * 1 * -18)) / (2 * 1) x = (-2 ± √(4 + 72)) / 2 x = (-2 ± √76) / 2 x = (-2 ± 2√19) / 2 x = -1 ± √19

Therefore, the two x-intercepts are x = -1 + √19 and x = -1 - √19.

Approximating the Roots: Locating the Intercepts

To determine the intervals between which the graph crosses the negative x-axis, we need to approximate the values of the x-intercepts. Since √19 is between √16 (which is 4) and √25 (which is 5), we can estimate √19 to be around 4.36. Therefore,

x₁ = -1 + √19 ≈ -1 + 4.36 ≈ 3.36 x₂ = -1 - √19 ≈ -1 - 4.36 ≈ -5.36

So, one x-intercept is approximately 3.36, which lies on the positive x-axis, and the other x-intercept is approximately -5.36, which lies on the negative x-axis. The question asks between which two ordered pairs the graph crosses the negative x-axis. The x-intercept we are interested in is -5.36. This value lies between -6 and -5.

Therefore, the graph of f(x) crosses the negative x-axis between the ordered pairs (-6, 0) and (-5, 0).

Conclusion: Mastering the Art of Intercept Identification

In this exploration, we have successfully navigated the process of finding the x-intercepts of a quadratic function, specifically f(x) = (1/2)x² + x - 9. We utilized the quadratic formula, a powerful tool that empowers us to solve any quadratic equation and uncover its roots. By approximating the roots, we pinpointed the intervals where the graph intersects the x-axis, specifically focusing on the negative x-axis. This comprehensive understanding of x-intercepts provides a solid foundation for analyzing and interpreting the behavior of quadratic functions and their applications in various mathematical and real-world contexts.

By mastering the techniques discussed in this guide, you can confidently tackle similar problems and gain a deeper appreciation for the elegance and utility of quadratic functions. Remember, practice is key to solidifying your understanding, so continue to explore different quadratic functions and challenge yourself to find their x-intercepts. With dedication and perseverance, you'll unlock the secrets of these mathematical marvels and expand your problem-solving prowess.

Answer

The graph of f(x) = (1/2)x² + x - 9 crosses the negative x-axis between the ordered pairs A. (-6, 0) and (-5, 0).