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Arithmetic Expressions - Quadratic Frog

[verano 2016 - Coralys - Ive]

Arithmetic expressions are an essential part of almost any algorithm that solves a useful problem. Therefore, a basic skill in any computer programming language is to implement arithmetic expressions correctly. In this laboratory experience you will practice the implementation of arithmetic expressions in C++ by writing equations for the quadratic formula and completing the code for a game in which a frog jumps from leaf to leaf.

Objectives:

Implement arithmetic expressions in C++ to produce the graphs that will define the trajectory of the frog’s jump in the game.
Use constant variables adequately.
Define variables using adequate data types.
Cast a data value to another type when necessary.

Pre-Lab:

Before you get to the laboratory you should have:

Reviewed the following concepts:

a. Implementing arithmetic expressions in C++.

b. Basic data types in C++ (int, float, double)

c. Using “type casting” to cast the value of variables to other data types within expressions.

d. Using arithmetic functions and constants from the cmath library.

e. Quadratic equations and their graphs.
Studied the concepts and instructions for the laboratory session.
Taken the Pre-Lab quiz available in Moodle.

Quadratic Formula

A quadratic equation has the form

$$y = a x^2+ b x + c,$$

where $$a, b, c$$ are real numbers and $$a\not=0$$. The graph of a quadratic equation is a parabola that opens up if $$a > 0$$ and opens down if $$a < 0$$.

A graph intersects the $$x$$-axis when $$ y = 0 $$ . Therefore, if a parabola intersects the $$x$$-axis, the intersects are given by the real solutions to the equation

$$0 = a x^2 + b x + c.$$

The solutions to the previous equation can be obtained using the quadratic formula:

$$x=\frac{-b±\sqrt{b^2-4ac}}{2a}.$$

Note that if the discriminant $$b^2-4ac$$ of the quadratic formula is negative, the values of $$x$$ are complex numbers and are not plotted in the Cartesian plane. Therefore, if the discriminant is negative, the parabola does not intersect the $$x$$-axis. If the discriminant is equal to $$0$$, then the parabola intersects the $$x$$-axis in only one point (only the vertex touches the $$x$$-axis).

If the discriminant is positive, the quadratic formula gives two solutions to the equation $$0 = a x^2 + b x + c$$ and these solutions are the intersects in the $$x$$-axis. For example, suppose that the quadratic formula gives two values

$$ x = x_1 $$ $$ x = x_2 $$

Then,

$$a x^2 + b x + c = a(x-x_1)(x-x_2),$$

where $$x_1$$ and $$x_2$$ are the intersects in the $$x$$-axis. If $$a<0$$, the graph of the parabola will be similar to the one in Figure 1.

Figure 1. Parabola that opens down and intersects the $$x$$-axis in $$x_1$$ and $$x_2$$.

Note that the equation

$$y=-(x-x_1)(x-x_2)$$

is a quadratic equation and its parabola opens down and intersects the $$x$$-axis in $$x_1$$ and $$x_2$$. For example, the equation

$$y=-(x+2)(x-3)=-x^2+x+6$$

is a quadratic equation with a parabola that opens down and intersects the $$x$$-axis at $$x_1=-2$$ and $$x_2=3$$. Note that, in this equation, the values for $$a, b, c$$ are $$a=-1, \ b=1, \ c=6$$.

!INCLUDE “../../eip-diagnostic/quadratic-frog/en/diag-quadratic-frog-01.html”

!INCLUDE “../../eip-diagnostic/quadratic-frog/en/diag-quadratic-frog-08.html”

!INCLUDE “../../eip-diagnostic/quadratic-frog/en/diag-quadratic-frog-09.html”

Laboratory session:

Exercise 1

In this exercise you will implement the quadratic formula to complete a game in which a frog leaps from one leaf to another. You will assume that the leaves are in the $$x$$-axis and that the leap is determined by a parabola that opens down. If you want the frog to leap from leaf to leaf, you must find a quadratic equation with a parabola that opens down and intersects the $$x$$-axis in the places where the leaves are located. Your task is to write the equations for the quadratic formula.

Instructions

Load the proyect QuadraticFrog into QtCreator. There are two ways to do this:
- Using the virtual machine: Double click the file QuadraticFrog.pro that can be found in the home/Documents/eip/Expressions-QuadraticFrog of your virtual machine.
- Downloading the proyect’s folder from Bitbucket: Use a terminal and write the command git clone http://bitbucket.org/eip-uprrp/expressions-quadraticfrog to download the folder QuadraticFrog from Bitbucket. Double click the file QuadraticFrog.prolocated in the folder that you downloaded to your computer.
Configure the project and run the program by clicking the green arrow in the menu on the left side of the Qt Creator window. The program will display a message saying that the quadratic formula is wrong. This happens because the program has testing instructions that verify that the code implementation is correct. Your program is missing the code for the quadratic formula and this is why the message is displayed.
You will write the equations for the quadratic formula in the file QuadraticFormula.cpp (in Sources). In the function QuadraticPlus add the equation

$$result=\frac{-b+\sqrt{b^2-4ac}}{2a},$$

and in the function QuadraticMinus add the equation

$$result=\frac{-b-\sqrt{b^2-4ac}}{2a}.$$

The other files of the project have code that will test your equations by evaluating several choices for $$a, b, c$$ and verifying that the equations produce the expected result. Code validation is an important part of software development.
Run the program by clicking the green arrow in the menu on the left side of the Qt Creator window. If your equations are implemented correctly, you should obtain a window similar to the one in Figure 2.

Figure 2. Window of the game Quadratic Frog.
Use “Deliver 1” in Moodle to submit the file QuadraticFormula.cpp containing the code with the functions QuadraticPlus and QuadraticMinus. Remember to use good programming practices, by including the names of the programmers and documenting your program.
To play, the frog should leap from one leaf to another. Note that the leaves have values for $$x_1$$ and $$x_2$$. These values represent the intersects of the parabola with the $$x$$-axis. You should input the values for the coefficients $$a, b, c$$ of the quadratic equation so that its graph that is a parabola that opens down and intersects the $$x$$-axis in the values $$x_1, x_2$$ shown in the leaves. You can obtain these values noting that

$$a x^2 + b x + c = a(x-x_1)(x-x_2),$$

as in the explanation above.

Exercise 2

In this exercise you will write a program to obtain a student’s grade point average (GPA).

Suppose that all courses in Yauco’s University are 3 credits each and have the following point values: $$A = 4$$ points per credit; $$B = 3$$ points per credit; $$C = 2$$ points per credit; $$D = 1$$ point per credit y $$F = 0$$ points per credit.