Quadratic Function Analysis C(x) = X² + 12x + 20 Vertex Intercepts And More

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In the realm of mathematics, quadratic functions hold a significant place, serving as versatile tools for modeling various real-world phenomena. Understanding the intricacies of these functions is crucial for students, educators, and anyone seeking to grasp the underlying principles governing parabolic relationships. In this comprehensive guide, we will delve into the analysis of a specific quadratic function, c(x) = x² + 12x + 20, exploring its various forms, key features, and applications.

(a) Transforming to Vertex Form: Unveiling the Parabola's Heart

The vertex form of a quadratic function, expressed as c(x) = a(x - h)² + k, provides valuable insights into the parabola's characteristics. The vertex, represented by the coordinates (h, k), marks the point where the parabola changes direction, either reaching its minimum or maximum value. The coefficient 'a' determines the parabola's concavity – whether it opens upwards (a > 0) or downwards (a < 0) – and its vertical stretch or compression.

To convert the given function, c(x) = x² + 12x + 20, into vertex form, we employ the technique of completing the square. This method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a squared binomial. Let's embark on this transformative journey step by step.

First, we focus on the quadratic and linear terms, x² + 12x. To complete the square, we need to add and subtract the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is 12, so half of it is 6, and its square is 36. Adding and subtracting 36, we get:

c(x) = x² + 12x + 36 - 36 + 20

Now, the first three terms, x² + 12x + 36, form a perfect square trinomial, which can be factored as (x + 6)². Substituting this back into the equation, we have:

c(x) = (x + 6)² - 36 + 20

Finally, combining the constant terms, -36 and 20, we arrive at the vertex form:

c(x) = (x + 6)² - 16

This transformation reveals the function's vertex form, where a = 1, h = -6, and k = -16. The positive value of 'a' indicates that the parabola opens upwards, signifying a minimum point at the vertex.

(b) Pinpointing the Vertex: The Parabola's Turning Point

The vertex of a parabola, as mentioned earlier, is the point where the curve changes direction. In the vertex form of the quadratic function, c(x) = a(x - h)² + k, the vertex is readily identified as the coordinates (h, k). For our function, c(x) = (x + 6)² - 16, we have already determined that h = -6 and k = -16.

Therefore, the vertex of the parabola represented by c(x) = x² + 12x + 20 is (-6, -16). This point marks the minimum value of the function, as the parabola opens upwards. The vertex serves as a crucial reference point for understanding the parabola's behavior and symmetry. The axis of symmetry, a vertical line passing through the vertex, divides the parabola into two symmetrical halves. In this case, the axis of symmetry is the line x = -6.

Understanding the vertex's significance is essential for various applications, such as optimization problems where we seek to find the maximum or minimum value of a function. The vertex provides the coordinates where this extreme value occurs, allowing us to solve real-world scenarios involving maximization or minimization.

(c) Unveiling the x-intercept(s): Where the Parabola Meets the Axis

The x-intercepts of a function are the points where the graph intersects the x-axis. At these points, the function's value, c(x), is equal to zero. To determine the x-intercepts of our function, c(x) = x² + 12x + 20, we need to solve the equation c(x) = 0.

x² + 12x + 20 = 0

This quadratic equation can be solved using various methods, including factoring, completing the square, or the quadratic formula. Let's explore the factoring method first. We need to find two numbers that add up to 12 (the coefficient of the linear term) and multiply to 20 (the constant term). The numbers 2 and 10 satisfy these conditions.

Therefore, we can factor the quadratic equation as follows:

(x + 2)(x + 10) = 0

For the product of two factors to be zero, at least one of them must be zero. Setting each factor equal to zero, we get:

x + 2 = 0 or x + 10 = 0

Solving for x in each equation, we find the x-intercepts:

x = -2 or x = -10

Thus, the x-intercepts of the parabola are (-2, 0) and (-10, 0). These points represent where the parabola crosses the x-axis. The x-intercepts provide valuable information about the function's behavior, indicating the points where the function's value changes sign.

(d) Discovering the y-intercept: Where the Parabola Embraces the Vertical Axis

The y-intercept of a function is the point where the graph intersects the y-axis. At this point, the x-coordinate is equal to zero. To determine the y-intercept of our function, c(x) = x² + 12x + 20, we need to evaluate the function at x = 0.

c(0) = (0)² + 12(0) + 20

c(0) = 20

Therefore, the y-intercept of the parabola is (0, 20). This point represents where the parabola crosses the y-axis. The y-intercept is a straightforward point to find, as it simply involves substituting x = 0 into the function's equation.

The y-intercept, along with the x-intercepts and vertex, provides a comprehensive understanding of the parabola's position and orientation in the coordinate plane. These key features allow us to sketch the graph of the function accurately and interpret its behavior.

(e) Discussion Category: Mathematics – Unveiling the Power of Quadratic Functions

The analysis of the quadratic function c(x) = x² + 12x + 20 falls squarely within the realm of mathematics, specifically within the subfield of algebra and calculus. Quadratic functions are fundamental building blocks in mathematical modeling, serving as representations for various real-world phenomena, including projectile motion, optimization problems, and curve fitting.

The exploration of the vertex form, intercepts, and other key features of quadratic functions equips us with a powerful toolkit for understanding and manipulating these mathematical entities. By mastering the techniques discussed in this guide, students and enthusiasts can confidently tackle a wide range of mathematical challenges involving quadratic functions.

The broader implications of quadratic functions extend beyond the classroom, permeating diverse fields such as physics, engineering, economics, and computer science. Their ability to model parabolic trajectories, optimize resource allocation, and approximate complex relationships makes them indispensable tools in various scientific and technological endeavors.

In conclusion, the analysis of the quadratic function c(x) = x² + 12x + 20 has provided a comprehensive exploration of its properties, from its vertex form to its intercepts. This understanding empowers us to appreciate the beauty and versatility of quadratic functions and their role in shaping our mathematical world.