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Ex2. Quadratic Functions in Standard Form

When Li is programming, his little sister comes, asking him about quadratic functions. While Li is a top coder, his knowledge in that part of mathematics is close to zero, so he turns to you for help. As a student taking VE280, you can use your knowledge about abstract data types (ADT) to help Li and his sister play with quadratic functions.

Related Topics: ADT.

Problem: Li wants to represent a quadratic function in a standard form, which is \(f(x) = ax^2+bx+c\) (\(a\neq 0\)). He decides that the following operations should be allowed on quadratic functions:

1. Evaluate \(f(x)\) at a given int \(x\) value.

2. Get the root(s) of \(f(x)\), which is the value of \(x\) such that \(f(x)=0\)

3. Check if two quadratic functions (\(f\) and \(g\)) intersects, which means whether there exists some real \(x\) such that \(f(x)=g(x)\).

Therefore, he designed this interface to represent a quadratic function

class QuadraticFunction {
    // OVERVIEW: the standard form of a quadratic function f(x) = ax^2 + bx + c
    float a;
    float b;
    float c;

public:
    QuadraticFunction(float a_in, float b_in, float c_in);
    // REQUIRES: a_in is nonezero
    // EFFECTS: creates a quadratic function in standard form

    float getA() const;
    // EFFECTS: returns the value of a

    float getB() const;
    // EFFECTS: returns the value of b

    float getC() const;
    // EFFECTS: returns the value of c

    float evaluate(float x);
    // EFFECTS: returns the value of f(x)

    Root getRoot();
    // EFFECTS: returns the roots of the quadratic function

    bool intersect(QuadraticFunction q);
    // EFFECTS: returns whether g and this intersect
};

Here, the constructor takes 3 inputs a_in, b_in and c_in and uses them to represent the quadratic function \(f(x)=ax^2+bx+c\). Also, the output function for this exercise is provided.

Requirements:

1. Look through file rootType.h, to make the output simple, we make the following restrictions:
   • if \(f(x)\) has two different real roots, then the smaller \(x_1\) should be in roots[0] and the bigger \(x_2\) should be in roots[1].
   • If \(f(x)\) has one real root, then \(x_1=x_2\) should be in both roots[0] and roots[1].
   • If \(f(x)\) has two complex roots, then \(x_1=m-ni\) should be in root[0] and \(x_2 = m+ni\) should be in roots[1], where \(n>0\).
2. Look through standardForm.h and implement the methods for QuadraticFunction class in standardForm.cpp.
3. ex2.cpp is used to test your ADT, you can just read it and run it.

Input Format: Since you only need to implement the methods of this ADT, we just provide a sample input. And there will not be cases where \(a=0\).

1 -3 2
1
2 -4 2

Output Format: Since you only need to implement the methods of this ADT, we just provide a sample output. NOTE that although in some cases it may be weird to have x1 = 1.0 + -1.0i, just ignore it.

f(x)=1.0x^2+-3.0x+2.0
f(1.0)=0.0
f(x) has 2 real roots.
x1 = 1.0 + 0.0i
x2 = 2.0 + 0.0i
1

Hint:

1. \(\Delta = b^2 - 4ac\), if \(\Delta \geq 0\), \(x=\frac{-b\pm \sqrt{\Delta}}{2a}\). Else, \(x=\frac{-b\pm i\sqrt{-\Delta}}{2a}\)

2. \(a\) for \(g(x)\) can be the same as \(a\) for \(f(x)\).