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LP_MC_FN_Nspire.java
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LP_MC_FN_Nspire.java
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// The "LP_NSpire" class.
import java.awt.*;
import hsa.Console;
import javax.imageio.*;
import java.io.*;
import java.awt.image.BufferedImage;
import java.text.*;
import java.util.*;
import java.lang.Object;
public class LP_MC_FN_NSpire
{
static Console c; // The output console
static char[] eq;
static double xmin, xmax, ymin, ymax, increment, ans;
static int[] xPixels, yPixels;
static double[] xValues, yValues;
static char choice;
static int lnCount, placeValue;
static boolean error;
//StatTab variables (linear)
public static ArrayList x = new ArrayList ();
public static ArrayList y = new ArrayList ();
public static double[] xArray;
public static double[] yArray;
public static double[] xyArray;
public static double[] xsqArray;
public static double[] ysqArray;
public static double xdoubleparse, ydoubleparse;
//StatTab variables (exponential)
public static double[] ylogArray;
//StatTab variables (quadratic)
public static double[] xcubeArray;
public static double[] xtesseractArray;
public static double[] xsqyArray;
public static double matrix0[] [] = new double [3] [4];
public static double matrix1[] [] = new double [3] [4];
public static double matrix2[] [] = new double [3] [4];
public static double matrix3[] [] = new double [3] [4];
public static double matrix4[] [] = new double [3] [4];
public static double matrix5[] [] = new double [3] [4];
public static double matrix6[] [] = new double [3] [4];
public static double[] sseArray;
public static double[] sstArray;
//****************************Parses and calculates an equation*****************************
//First major method: converts a char array (the equation) into a string array which works better
public static String[] makeInput (char[] eq, double x)
{
String[] input = new String [eq.length];
int count = 0;
try
{
//Goes through all the chars
for (int i = 0 ; i < eq.length ; i++)
{
//Groups numbers together (e.g. 2,3,5,.,7 (5 separate elements) becomes 235.7 (one element)
if (java.lang.Character.isDigit (eq [i]) == true || eq [i] == '.' && java.lang.Character.isDigit (eq [i - 1]) == false && eq [i - 1] != '.')
{
String num = "";
while (java.lang.Character.isDigit (eq [i]) == true || eq [i] == '.')
{
num = num + eq [i];
i++;
if (i == eq.length)
{
break;
}
}
i--;
input [count] = num;
count++;
}
//Groups functions together (e.g. s,i,n (3 separate elements) becomes sin (one element)
else if (eq [i] == 's' && eq [i + 1] == 'i' && eq [i + 2] == 'n')
{
i = i + 2;
input [count] = "sin";
count++;
}
else if (eq [i] == 'c' && eq [i + 1] == 'o' && eq [i + 2] == 's')
{
i = i + 2;
input [count] = "cos";
count++;
}
else if (eq [i] == 't' && eq [i + 1] == 'a' && eq [i + 2] == 'n')
{
i = i + 2;
input [count] = "tan";
count++;
}
else if (eq [i] == 'l' && eq [i + 1] == 'n')
{
i = i++;
input [count] = "ln";
count++;
}
else if (eq [i] == 's' && eq [i + 1] == 'q' && eq [i + 2] == 'r' && eq [i + 3] == 't')
{
i = i + 3;
input [count] = "sqrt";
count++;
}
else if (eq [i] == 'n' && eq [i + 1] == 'e' && eq [i + 2] == 'g')
{
i = i + 2;
input [count] = "neg";
count++;
}
//Replaces expressions like x or pi with their numerical values
else if (eq [i] == 'x')
{
input [count] = Double.toString (x);
count++;
}
else if (eq [i] == 'p' && eq [i + 1] == 'i')
{
i = i + 1;
input [count] = String.valueOf (3.1415926535897932);
count++;
}
else if (eq [i] == 'e')
{
input [count] = String.valueOf (2.7182818284590452);
count++;
}
else if (eq [i] == 'a' && eq [i + 1] == 'n' && eq [i + 2] == 's')
{
i = i + 2;
input [count] = String.valueOf (ans);
count++;
}
//Everything else is left untouched (e.g. +, -)
else
{
input [count] = Character.toString (eq [i]);
count++;
}
}
}
//Identifies indexoutofbounds exceptions (which are generated from invalid equations)
catch (IndexOutOfBoundsException e)
{
error = true;
}
finally
{
return input;
}
}
//Second major method: converts string array from makeInput into reverse Polish notation using the shunting yard algorithm
//http://en.wikipedia.org/wiki/Shunting-yard_algorithm#The_algorithm_in_detail
//Essentially I followed the instructions from the above link
//The following comments are mostly copy-pasted from the link
public static String[] reversePolish (String[] input)
{
String[] output = new String [input.length];
String[] operator = new String [input.length];
int outputCount = 0;
int operatorCount = 0;
//Read a token
for (int i = 0 ; i < input.length ; i++)
{
if (input [i] != null && input [i] != "")
{
//If the token is a number, then add it to the output queue
if (identifier (input [i]) == "number")
{
output [outputCount] = input [i];
outputCount++;
}
//If the token is a function token, then push it onto the stack
else if (identifier (input [i]) == "function")
{
operator [operatorCount] = input [i];
operatorCount++;
}
//If the token is an operator o1
else if (identifier (input [i]) == "operator")
{
//push o1 onto the stack
if (operatorCount == 0)
{
operator [operatorCount] = input [i];
operatorCount++;
}
//while there is an operator token, o2, at the top of the stack
else
{
//o1 is left-associative and its precedence is *less than or equal* to that of o2
if (input [i].equals ("^") == false)
{
while (operatorCount != 0 && orderOps (operator [operatorCount - 1]) >= orderOps (input [i]))
{
output [outputCount] = operator [operatorCount - 1];
outputCount++;
operator [operatorCount - 1] = null;
operatorCount--;
}
//push o1 onto the stack
operator [operatorCount] = input [i];
operatorCount++;
}
//o1 if right associative, and has precedence *less than* that of o2
else
{
while (operatorCount != 0 && orderOps (operator [operatorCount - 1]) > orderOps (input [i]))
{
output [outputCount] = operator [operatorCount - 1];
outputCount++;
operator [operatorCount - 1] = null;
operatorCount--;
}
//push o1 onto the stack
operator [operatorCount] = input [i];
operatorCount++;
}
}
}
else if (identifier (input [i]) == "bracket")
{
//If the token is a left parenthesis, then push it onto the stack
if (input [i].equals ("(") == true)
{
operator [operatorCount] = input [i];
operatorCount++;
}
//If the token is a right parenthesis
else
{
//Until the token at the top of the stack is a left parenthesis, pop operators and functions off the stack onto the output queue
while (operatorCount != 0 && operator [operatorCount - 1].equals ("(") == false)
{
output [outputCount] = operator [operatorCount - 1];
outputCount++;
operator [operatorCount - 1] = null;
operatorCount--;
}
//Pop the left parenthesis from the stack, but not onto the output queue
if (operatorCount != 0 && operator [operatorCount - 1].equals ("(") == true)
{
operator [operatorCount - 1] = null;
operatorCount--;
}
}
}
}
}
//When there are no more tokens to read
//While there are still operator tokens in the stack
for (operatorCount = operatorCount ; operatorCount > 0 ; operatorCount--)
{
//Pop the operator onto the output queue
output [outputCount] = operator [operatorCount - 1];
outputCount++;
operator [operatorCount - 1] = null;
}
return output;
}
//Third major method: evaluates the reverse Polish notation generated by reversePolish
//http://en.wikipedia.org/wiki/Reverse_Polish_notation#Postfix_algorithm
//Essentially I followed the instructions from the above link
//The following comments are mostly copy-pasted from the link
public static double evaluate (String[] output)
{
double[] result = new double [output.length];
int count = 0;
try
{
//Read the next token from input
for (int i = 0 ; i < output.length ; i++)
{
if (output [i] != null)
{
//If the token is a value
if (identifier (output [i]) == "number")
{
//Push it onto the stack
result [count] = Double.parseDouble (output [i]);
count++;
}
//If the token is a function (i.e. an operation with one argument)
//Pop the top value
//Evaluate the operator on the top value
//Push the returned result back onto the stack
else if (identifier (output [i]) == "function")
{
if (output [i].equals ("sin") == true)
{
result [count - 1] = Math.sin (result [count - 1]);
}
else if (output [i].equals ("cos") == true)
{
result [count - 1] = Math.cos (result [count - 1]);
}
else if (output [i].equals ("tan") == true)
{
result [count - 1] = Math.tan (result [count - 1]);
}
else if (output [i].equals ("ln") == true)
{
result [count - 1] = Math.log (result [count - 1]);
}
else if (output [i].equals ("sqrt") == true)
{
result [count - 1] = Math.sqrt (result [count - 1]);
}
else if (output [i].equals ("neg") == true)
{
result [count - 1] = -(result [count - 1]);
}
}
//If the token is an operation (i.e. has two arguments)
//Pop the top two values
//Evaluate the operator on the top two values
//Push the returned result back onto the stack
else if (identifier (output [i]) == "operator")
{
if (output [i].equals ("+") == true)
{
result [count - 2] = result [count - 1] + result [count - 2];
count--;
}
else if (output [i].equals ("-") == true)
{
result [count - 2] = result [count - 2] - result [count - 1];
count--;
}
else if (output [i].equals ("*") == true)
{
result [count - 2] = result [count - 1] * result [count - 2];
count--;
}
else if (output [i].equals ("/") == true)
{
result [count - 2] = result [count - 2] / result [count - 1];
count--;
}
else if (output [i].equals ("^") == true)
{
result [count - 2] = Math.pow (result [count - 2], result [count - 1]);
count--;
}
}
}
}
}
//Identifies indexoutofbounds exceptions (which are generated from invalid equations)
catch (IndexOutOfBoundsException e)
{
error = true;
}
finally
{
//If there is only one value in the stack, that value is the result of the calculation
ans = result [0];
return result [0];
}
}
//Sub method - order of operations
public static int orderOps (String op)
{
if (op.equals ("+") == true)
{
return 1;
}
else if (op.equals ("-") == true)
{
return 1;
}
else if (op.equals ("/") == true)
{
return 2;
}
else if (op.equals ("*") == true)
{
return 2;
}
else if (op.equals ("^") == true)
{
return 3;
}
else if (identifier (op).equals ("function"))
{
return 4;
}
else
{
return 0;
}
}
//Sub method - identifies a string
public static String identifier (String term)
{
if (
term.equals ("sin") == true ||
term.equals ("cos") == true ||
term.equals ("tan") == true ||
term.equals ("ln") == true ||
term.equals ("sqrt") == true ||
term.equals ("neg") == true
)
{
return "function";
}
else if (
term.equals ("+") == true ||
term.equals ("-") == true ||
term.equals ("*") == true ||
term.equals ("/") == true ||
term.equals ("^") == true
)
{
return "operator";
}
else if (
term.startsWith ("1") == true ||
term.startsWith ("2") == true ||
term.startsWith ("3") == true ||
term.startsWith ("4") == true ||
term.startsWith ("5") == true ||
term.startsWith ("6") == true ||
term.startsWith ("7") == true ||
term.startsWith ("8") == true ||
term.startsWith ("9") == true ||
term.startsWith ("0") == true ||
term.startsWith (".") == true ||
term.startsWith ("-") == true
)
{
return "number";
}
else if (
term.equals ("(") == true ||
term.equals (")") == true
)
{
return "bracket";
}
else
{
return "error";
}
}
//Groups all three major methods together
public static double calculate (char[] message, double x)
{
String[] input = makeInput (message, x);
String[] output = reversePolish (input);
return evaluate (output);
}
//Displays coordinates
public static void displayC (double x, double y)
{
String output = "(" + x + "," + y + ")";
c.setColor (Color.black);
Font f = new Font ("Arial", Font.PLAIN, 12);
c.setFont (f);
//Ensures that the coordinates are placed properly (so that it appears fully on the screen)
int xshift = -1;
int yshift = 1;
if (xPixel (x) > 320)
{
xshift = 1;
}
if (yPixel (y) < 300)
{
yshift = -1;
}
c.drawString (output, xPixel (x) + xshift * 5, yPixel (y) - yshift * 5);
}
//This method is used solely to ensure that the graph window makes sense
//i.e. xmin<xmax, ymin<ymax
public static double checkRange (double a, String prompt)
{
print (prompt);
double z = readDouble ();
for (int i = 0 ;; i++)
{
if (z >= a)
{
break;
}
else
{
c.print ("Not in range - must be greater than " + a + ". ");
print (prompt);
z = readDouble ();
}
}
return z;
}
//This method is used to ensure that inputs for certain functions are within the graph window
//The graph window is defined by xmin, xmax, ymin, and ymax
public static double filter (String prompt)
{
print (prompt);
double z = readDouble ();
for (int i = 0 ;; i++)
{
if (z >= xmin && z <= xmax)
{
break;
}
else
{
c.print ("Not in graph range. ");
print (prompt);
z = readDouble ();
}
}
return z;
}
//Order of magnitude method - used for axes
//e.g. if the graph window is 100 wide, each axis tick is 10
public static double OOM (double x)
{
x = Math.abs (x);
double OOM = 0; //to avoid errors
if (x < 1 && x != 0)
{
x = 1 / x;
for (int i = 0 ;; i++)
{
if (Math.pow (10, i) < x && Math.pow (10, i + 1) > x)
{
OOM = Math.pow (10, -i - 1);
break;
}
else if (Math.pow (10, i) == x)
{
OOM = Math.pow (10, -i - 1);
break;
}
}
}
else
{
for (int i = 0 ;; i++)
{
if (Math.pow (10, i) < x && Math.pow (10, i + 1) > x)
{
OOM = Math.pow (10, i);
break;
}
else if (Math.pow (10, i) == x)
{
OOM = Math.pow (10, i - 1);
break;
}
}
}
return OOM;
}
//This method makes sure everything fits in the dialogue box (which is two lines high)
public static void print (String message)
{
c.println (message);
lnCount++;
//Resets the dialogue box
if (lnCount == 2)
{
c.setCursor (1, 1);
lnCount = 0;
}
}
//Similar to the print method
public static int readInt ()
{
int output = c.readInt ();
lnCount++;
if (lnCount == 2)
{
c.setCursor (1, 1);
lnCount = 0;
}
return output;
}
//Similar to the print method
public static double readDouble ()
{
double output = c.readDouble ();
lnCount++;
if (lnCount == 2)
{
c.setCursor (1, 1);
lnCount = 0;
}
return output;
}
//Similar to the print method
public static String readString ()
{
String output = c.readLine ();
lnCount++;
if (lnCount == 2)
{
c.setCursor (1, 1);
lnCount = 0;
}
return output;
}
//Displays the graph axes
//Output is based on xmin, xmax, ymin, ymax (i.e. the graph window)
public static void axes ()
{
//Draws the x and y axis lines
c.setColor (Color.black);
int xC = (int) (-(xmin * 640) / (xmax - xmin));
c.drawLine (xC, 100, xC, 500);
int yC = 100 + (int) (-(ymax * 400) / (ymin - ymax));
c.drawLine (0, yC, 640, yC);
//Ticks
int OOMX = xPixel (OOM ((xmax - xmin) / 2)) - xPixel (0);
int OOMY = -yPixel (OOM ((ymax - ymin) / 2)) + yPixel (0);
c.setColor (Color.black);
Font f = new Font ("Arial", Font.PLAIN, 12);
c.setFont (f);
//Calculates an appropriate tick length
String numX = Double.toString (OOM ((xmax - xmin) / 2));
String numY = Double.toString (OOM ((ymax - ymin) / 2));
//Positive x
for (int i = xPixel (0) ; i < 640 ; i = i + OOMX)
{
c.drawLine (i, yPixel (0) - 5, i, yPixel (0) + 5);
if (i == xPixel (0) + OOMX)
{
c.drawString (numX, i - 5, yPixel (0) + 20);
}
}
//Negative x
for (int i = xPixel (0) ; i > 0 ; i = i - OOMX)
{
c.drawLine (i, yPixel (0) - 5, i, yPixel (0) + 5);
}
//Positive y
for (int i = yPixel (0) ; i > 100 ; i = i - OOMY)
{
c.drawLine (xPixel (0) - 5, i, xPixel (0) + 5, i);
if (i == yPixel (0) - OOMY)
{
c.drawString (numY, xPixel (0) + 10, i + 5);
}
}
//Negative y
for (int i = yPixel (0) ; i < 500 ; i = i + OOMY)
{
c.drawLine (xPixel (0) - 5, i, xPixel (0) + 5, i);
}
}
//*********************************Four crucial methods*****************************
//Converts a number x into the pixel location in the console
public static int xPixel (double x)
{
int xC = (int) (((x - xmin) * 640) / (xmax - xmin));
return xC;
}
//Converts a number y into the pixel location in the console
public static int yPixel (double y)
{
int yC = 100 + (int) (((y - ymax) * 400) / (ymin - ymax));
return yC;
}
//Converts a pixel location in the console into a number x
public static double reverseX (int pixel)
{
double x = pixel * (xmax - xmin) / 640 + xmin;
return x;
}
//Converts a pixel location in the console into a number y
public static double reverseY (int pixel)
{
double y = (pixel - 100) * (ymin - ymax) / 400 + ymax;
return y;
}
//Loads image from file
public static Image loadImage (String name)
{
Image img = null;
try
{
img = ImageIO.read (new File (name));
}
catch (IOException e)
{
}
return img;
}
//The derivative function (the slope at a point)
public static void derivative (double[] xValues, double[] yValues, double a)
{
double output = 0; //to avoid errors
int k = 0; //to avoid errors
//Finds the approximate derivative value
for (int i = 0 ; i < xValues.length ; i++)
{
if (xValues [i] < a && xValues [i + 1] > a)
{
//This expression computes slope by comparing the two stored values closest to the input a
output = -(yValues [i] - yValues [i + 1]) / increment;
k = i;
}
}
//Prints result
c.print ("The derivative at ");
c.print (a, 0, 2);
c.print (" is ");
c.println (output, 0, 2);
//Displays the derivative line
displayC (a, (yValues [k] + yValues [k + 1]) / 2);
int newA = xPixel (a);
int newX = xPixel ((xValues [k] + xValues [k + 1]) / 2);
int newY = yPixel ((yValues [k] + yValues [k + 1]) / 2);
double b = reverseY (newY) - output * reverseX (newX);
//Graphs the line using a process similar to the standard graph method
for (double x = xmin ; x <= xmax ; x = x + increment)
{
double y = x * output + b;
int xC = xPixel (x);
int yC = yPixel (y);
if (yC > 100)
{
c.setColor (Color.red);
c.drawLine (xC, yC, xPixel (x + increment), yPixel (output * (x + increment) + b));
}
}
}
//The zeros function
public static void zeros (double[] xValues, double[] yValues, double a, double b)
{
double output = 0; //to avoid errors
//Finds the stored y value closest to zero between a and b
for (int i = 0 ; i < xValues.length - 1 ; i++)
{
if (xValues [i] >= a && xValues [i] <= b)
{
if (yValues [i] < 0 && yValues [i + 1] > 0 || yValues [i] > 0 && yValues [i + 1] < 0)
{
output = -(yValues [i] + yValues [i + 1]) / 2;
}
else if (yValues [i] == 0)
{
output = yValues [i];
}
}
}
//Prints the result
c.print ("The zero is ");
c.println (output, 0, 2);
//Draws the zero point
int outputPixel = xPixel (output);
Color col = new Color (139, 69, 19);
c.setColor (col);
c.fillOval (outputPixel - 5, yPixel (0) - 5, 10, 10);
displayC (output, 0);
}
//The maximum function
public static void max (double[] xValues, double[] yValues, double a, double b)
{
double output = yValues [0];
int x = xPixel (xValues [0]); //to avoid errors
//Finds the stored y value which is the largest within range
for (int i = 0 ; i < yValues.length ; i++)
{
if (yValues [i] > output && xValues [i] <= b && xValues [i] >= a)
{
output = yValues [i];
x = xPixel (xValues [i]);
}
}
//Prints the result
c.print ("The maximum from ");
c.print (a, 0, 2);
c.print (" to ");
c.print (b, 0, 2);
c.print (" is ");
c.println (output, 0, 2);
//Draws the maximum point
c.setColor (Color.green);
int outputPixel = yPixel (output);
c.fillOval (x - 5, outputPixel - 5, 10, 10);
displayC (x, output);
}
//The minimum function
public static void min (double[] xValues, double[] yValues, double a, double b)
{
double output = yValues [0];
int x = xPixel (xValues [0]); //to avoid errors
//Finds the stored y value which is the smallest within range
for (int i = 0 ; i < yValues.length ; i++)
{
if (yValues [i] < output && xValues [i] <= b && xValues [i] >= a)
{
output = yValues [i];
x = xPixel (xValues [i]);
}
}
//Prints the result
c.print ("The minimum from ");
c.print (a, 0, 2);
c.print (" to ");
c.print (b, 0, 2);
c.print (" is ");
c.println (output, 0, 2);
//Draws the minimum point
c.setColor (Color.blue);
int outputPixel = yPixel (output);
c.fillOval (x - 5, outputPixel - 5, 10, 10);
displayC (x, output);
}
//Load screen - purely for aesthetic purposes
//Mimics the actual TI-NSpire load screen
public static void loadScreen ()
{
//Checkerboard pattern
for (int i = 0 ; i < 32 ; i++)
{
for (int j = 0 ; j < 25 ; j++)
{
if ((i + j) % 2 == 0)
{
c.setColor (Color.red);
c.fillRect (i * 20, j * 20, 20, 20);
}
else
{
c.setColor (Color.black);
c.fillRect (i * 20, j * 20, 20, 20);
}
}
}
//The arc-like area at the bottom of the screen
c.setColor (Color.blue);
for (int i = 1 ; i <= 400 ; i++)
{
c.drawArc (-200, -200 + i, 900, 600, 180, 180);
}
c.setColor (Color.white);
c.fillRect (215, 235, 210, 30);
//Displays calculator name
Font f = new Font ("Courier", Font.BOLD, 60);
c.setFont (f);
c.drawString ("FML - NSpire", 100, 140);
//Fills the loading bar
c.setColor (Color.black);
for (int x = 0 ; x <= 200 ; x++)
{
c.fillRect (220, 240, x, 20);
try
{
Thread.currentThread ().sleep (20);
}
catch (Exception e)
{
}
}
}
//Displays the dialogue box
public static void box ()
{
c.setColor (Color.gray);
c.fillRect (0, 0, 550, 50);
c.setColor (Color.white);
c.fillRect (0, 0, 540, 40);
}
//Calculates the integral function (the area between the curve and the x axis)
public static void integral (double[] xValues, double[] yValues, double a, double b)
{
double sum = 0;
//Sums all the y values in range
for (int i = 0 ; i < xValues.length ; i++)
{
if (xValues [i] >= a && xValues [i] <= b)
{
sum = sum + yValues [i];
}
}
//Multiplies by the increment to get area
sum = sum * increment;
//Prints the result
c.print ("The integral from ");
c.print (a, 0, 2);
c.print (" to ");
c.print (b, 0, 2);
c.print (" is ");
c.println (sum, 0, 2);
}
//Displays the integral as the area between the curve and the x axis
public static void displayIntegral (int[] xPixels, int[] yPixels, double a, double b)
{
Color col = new Color (133, 133, 133);
c.setColor (col);
int zero = 100 + (int) (-(ymax * 400) / (ymin - ymax));
int newA = xPixel (a);
int newB = xPixel (b);
//Draws a series of lines from each y value to the point on the x axis directly below it
//Since each line is so close, it looks like a filled area
for (int i = 0 ; i < xPixels.length ; i++)
{
if (xPixels [i] >= newA && xPixels [i] <= newB)
{
c.drawLine (xPixels [i], zero, xPixels [i], yPixels [i]);
}
}
}
//Graph method - very important
public static void graph (char[] eq)
{
//Public arrays which store all the x and y values which are graphed, as well as their pixel values
xPixels = new int [(int) (Math.round ((xmax - xmin) / increment + 1))];
yPixels = new int [(int) (Math.round ((xmax - xmin) / increment + 1))];
xValues = new double [(int) (Math.round ((xmax - xmin) / increment + 1))];
yValues = new double [(int) (Math.round ((xmax - xmin) / increment + 1))];
int count = 0;
//This section does the graphing
//Increment is very small (about 1/2000 of the graph window)
for (double x = xmin ; x <= xmax ; x = x + increment)
{
//Calculates y given x
double y = calculate (eq, x);
int xC = xPixel (x);
int yC = yPixel (y);
//Stores values
xValues [count] = x;
yValues [count] = y;
xPixels [count] = xC;
yPixels [count] = yC;
count++;
//Draws a very short line between this coordinate and the next one
//The lines are short enough that the final result looks like a curve
c.setColor (Color.black);
if (yC >= 100)
{
c.drawLine (xC, yC, xPixel (x + increment), yPixel (calculate (eq, x + increment)));
}
}
}
//Trace function - given an x value, outputs the y value
public static void trace (double a)
{
//Calculates the y value