Analyzing The Quadratic Function P(x) = X(x-1) + 1
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In this comprehensive article, we will delve deep into the analysis of the quadratic function p(x) = x(x-1) + 1. Our primary goal is to dissect this function, revealing its key characteristics and properties. Specifically, we aim to determine the values of the coefficients a, b, and c when the function is expressed in its standard quadratic form, which is ax² + bx + c. Furthermore, we will calculate the discriminant of this quadratic function, a critical value that provides insights into the nature and number of its roots. By meticulously examining these aspects, we will gain a thorough understanding of the behavior and graphical representation of the given quadratic function. This exploration will not only enhance our grasp of quadratic functions in general but also equip us with the tools to analyze and interpret similar functions effectively. The value of understanding quadratic functions extends beyond the classroom, as they are fundamental in various fields, including physics, engineering, economics, and computer science. For instance, they can model projectile motion, optimize business processes, and design efficient algorithms. Therefore, mastering the concepts related to quadratic functions is a valuable asset for anyone pursuing studies or a career in these areas. Our journey begins with expanding and simplifying the given function to bring it into the standard quadratic form. This transformation is crucial because the coefficients a, b, and c are readily identifiable in this form. Once we have these values, we can easily compute the discriminant using the standard formula. The discriminant, in turn, will tell us whether the function has two distinct real roots, one repeated real root, or no real roots. This information is vital for sketching the graph of the function and understanding its behavior. Throughout this article, we will provide clear explanations and step-by-step calculations to ensure that the concepts are easily understood. We will also use illustrative examples and diagrams to further clarify the key ideas. By the end of this exploration, you will have a solid foundation in analyzing quadratic functions and be able to confidently tackle problems involving them.
Transforming p(x) into Standard Quadratic Form
To begin our analysis, we must first transform the given function, p(x) = x(x-1) + 1, into its standard quadratic form, which is ax² + bx + c. This form is essential because it allows us to easily identify the coefficients a, b, and c, which are crucial for further analysis, including calculating the discriminant and finding the roots of the function. The process of transformation involves expanding the expression and then simplifying it by combining like terms. This is a fundamental algebraic technique that is widely used in various mathematical contexts. In our case, we start by distributing the x in the term x(x-1). This means multiplying x by both x and -1. This step is based on the distributive property of multiplication over addition, which states that a(b + c) = ab + ac. Applying this property, we get x(x-1) = xx - x1 = x² - x. The next step is to substitute this back into the original function. So, p(x) = x(x-1) + 1 becomes p(x) = x² - x + 1. Now, we have the function in a simplified form, but it's essential to recognize that it is already in the standard quadratic form. The standard form ax² + bx + c represents a polynomial of degree two, where a, b, and c are constants. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. In our transformed function, p(x) = x² - x + 1, we can clearly see the correspondence with the standard form. The coefficient of the x² term is a, the coefficient of the x term is b, and the constant term is c. By comparing the transformed function with the standard form, we can directly identify the values of a, b, and c. This is a significant step in our analysis because these values are the building blocks for further calculations and interpretations. The identification of a, b, and c is a cornerstone of understanding quadratic functions, as they dictate the shape, position, and behavior of the parabola represented by the function. For example, the sign of a determines whether the parabola opens upwards (if a is positive) or downwards (if a is negative). The values of b and c influence the position of the vertex and the y-intercept of the parabola, respectively. Therefore, a thorough understanding of how to transform a quadratic function into its standard form and identify the coefficients is essential for anyone studying mathematics, physics, or engineering. In the subsequent sections, we will use these values to calculate the discriminant and explore its implications for the roots of the function. We will also discuss how these parameters affect the graphical representation of the quadratic function. By mastering these concepts, you will be well-equipped to analyze and interpret a wide range of quadratic functions.
Determining the Values of a, b, and c
Having successfully transformed the quadratic function p(x) = x(x-1) + 1 into its standard form, p(x) = x² - x + 1, we are now poised to identify the crucial coefficients a, b, and c. These coefficients are the bedrock upon which our subsequent analysis will be built, providing the necessary parameters for calculating the discriminant and understanding the function's behavior. The coefficient a is defined as the numerical factor multiplying the x² term. In our standard form p(x) = x² - x + 1, the x² term is simply x², which can be interpreted as 1x²*. Therefore, the value of a is 1. This might seem like a straightforward observation, but it carries significant implications. A positive value for a indicates that the parabola representing the function opens upwards, a fundamental characteristic that shapes the graph and its properties. Conversely, a negative value for a would imply a downward-opening parabola. The magnitude of a also affects the
