Question

Consider a quadratic function \(f(x ; a, b, c)=a x^{2}+b x+c .\) Make a class Quadratic for representing \(f\), where \(a, b\), and \(c\) are attributes, and the methods are 1\. value for computing a value of \(f\) at a point \(x\), 2\. table for writing out a table of \(x\) and \(f\) values for \(n x\) values in the interval \([L, R]\) 3\. roots for computing the two roots. Name of program file: Quadratic.py.

Short answer

Define class `Quadratic` with methods for value, table, and roots in Quadratic.py.

Step-by-step solution

Define the Class and Initialize Attributes

Begin by defining the class `Quadratic` in the file `Quadratic.py`. Create an `__init__` method to initialize the attributes `a`, `b`, and `c`, which represent the coefficients of the quadratic function.

Implement the value method

Within the `Quadratic` class, define a method named `value(self, x)`. In this method, compute the value of the quadratic function at a given `x` using the formula: \[ f(x) = ax^2 + bx + c \] Return the computed value.

Implement the table method

Define a method named `table(self, L, R, n)`. Calculate the step size using \( \text{step} = \frac{R - L}{n} \). Loop from `L` to `R` with this step size, and for each \( x \), compute \( f(x) \) using the `value` method and print or store these values in a tabular format.

Implement the roots method

Define a method named `roots(self)`. Use the quadratic formula to find the roots: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Calculate the discriminant \( b^2 - 4ac \). If it is negative, return a message indicating no real roots. Otherwise, compute the two roots and return them.

Key concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be represented in the standard form: \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The shape of its graph is a parabola which can open upwards or downwards depending on the sign of \(a\).

Here are some key properties of quadratic functions:

• **Vertex**: The vertex of a parabola is its highest or lowest point, depending on whether it opens downward or upward. It can be found using the formula \(x = -\frac{b}{2a}\).
• **Axis of symmetry**: The line \(x = -\frac{b}{2a}\) divides the parabola into two mirror images and passes through the vertex.
• **Roots**: These are the values of \(x\) where the function \(f(x) = 0\). They can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• **Discriminant**: Part of the quadratic formula, \(b^2 - 4ac\), which determines the nature of the roots. If it's positive, there are two distinct real roots; if zero, one real root; and if negative, no real roots.

Object-Oriented Programming

Object-oriented programming (OOP) is a programming paradigm that uses objects and classes to structure and organize software design. OOP principles help programmers manage code complexity by creating reusable and modular components.

In the context of defining a quadratic function in Python, OOP principles can be applied using classes and methods:

• **Class**: A blueprint for creating objects (instances), encapsulating data and behaviors that belong together. In our exercise, the class `Quadratic` includes attributes \(a\), \(b\), and \(c\).
• **Attributes**: Variables that hold data specific to the object. For the `Quadratic` class, they store the coefficients of the quadratic function.
• **Methods**: Functions defined inside a class that can manipulate object data. The methods `value`, `table`, and `roots` in the `Quadratic` class perform operations on the quadratic function.
• **Encapsulation**: The concept of bundling data with methods operating on that data. This simplifies code maintenance and enhances data security by restricting direct access to class attributes.

Practicing OOP enables efficient code reusability and easier troubleshooting, especially in complex programs.

Mathematical Methods in Python

Python offers various tools and libraries to perform mathematical operations, making it an excellent choice for implementing mathematical concepts such as quadratic functions.

In the Quadratic class, you can utilize the following tools:

• **Arithmetic Operations**: Basic operations like addition, subtraction, multiplication, and exponentiation to compute the quadratic function value.
• **Math Module**: Python’s built-in library providing advanced mathematical functions. In the `roots` method of the `Quadratic` class, \(\text{math.sqrt()}\) can calculate the square root, enabling us to solve the quadratic equation.
• **Loops and Conditionals**: Essential control structures in Python. For example, a loop is used in the `table` method to iterate over a range of x-values. Conditional statements evaluate whether the discriminant is non-negative to ensure real-valued roots.

By employing these methods, programmers can seamlessly blend mathematical formulas and coding structures, facilitating the implementation of efficient and functional programs.