Plotting equations on a cartesian Plane

some general:

GraphingPanel p = new GraphingPanel();
    JPanel[] Panel = new JPanel[2];
    JLabel[] Label = new JLabel[100];
    JTextField[] Field = new JTextField[100];
    JButton[] Enter = new JButton[100];
    JButton[] Clear = new JButton[100];
    JComboBox Subjects = new JComboBox();
    JComboBox Topics = new JComboBox();

why do you declare such big fields of components??
Please name them variables starting with lower case, otherwise you will get in trouble with object names.

To your question:
You need to repaint() your JPanel.

EDIT:

// Let's start with instances of what we really need:
// I'm using hungarian notation, which is one way to get a clear variable naming. 
// the "o" in front stands for "Object".
 
// by the way: this is a comment, you should use some to make clear what you do.
    GraphingPanel oGraphPanel = new GraphingPanel();
    JPanel oPanel = new JPanel();
    JLabel oLabel = new JLabel();
    JTextField oField = new JTextField();
    JButton oEnter = new JButton();
    JButton oClear = new JButton();
    JComboBox oSubjects = new JComboBox();
    JComboBox oTopics = new JComboBox();


/*
 * Ok, we've got our Components ready and now 
 * you should change your code to fit to this.
 */

// {YOUR CODE of class Graphs }

// to make it easier, I will add a methode for updating the GraphingPanel:
private void updateGraphPanel(){ // always make sure the component is there and initalized
    if(null != oGraphPanel){
        oGraphPanel.setText(oField.getText()); // passing the text from Graphs to GraphingPanel
        oGraphPanel.repaint(); // this triggers the paint method of GraphinPanel to be carried out again
    }
}

// GraphingPanel
public class GraphingPanel extends JPanel
{
    private String strText=null; // init with null (=not valid)

    public void setText(String strText){
        this.strText = strText;
    }

// {Your code of GraphingPanel}


}

That's about all. the Text is passed to GraphingPanel and then the repaint is triggered.
The text is not yet used there - depends on what you want to do with it. But it's available - so have fun.

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EDIT2:
Here is a jump start. compare this to your code (eclipse: right click in Package Explorer -> Compare with...each other).

import java.awt.BasicStroke;
import java.awt.Color;
import java.awt.Font;
import java.awt.Graphics;
import java.awt.Graphics2D;
import java.awt.geom.Line2D;

import javax.swing.BorderFactory;
import javax.swing.JPanel;
 
public class GraphingPanel extends JPanel
{
    public GraphingPanel()
    {
        setBounds(50, 50, 500, 400);
        setBorder(BorderFactory.createLineBorder(Color.BLACK, 2));
        setBackground(Color.WHITE);
    }
 
    @Override
	public void paintComponent(Graphics g)
    {
        super.paintComponents(g);
 
        int panelWidth = getWidth();
        int panelHeight = getHeight();
 
        g.setColor(Color.WHITE);
        g.fillRect(0, 0, panelWidth, panelHeight);
        g.setColor(Color.BLACK);
 
        Graphics2D g2 = (Graphics2D) g;
        g2.setStroke(new BasicStroke(1f));
 

        g2.draw(new Line2D.Double( 0, panelHeight/2, panelWidth ,panelHeight/2 ));
        g2.draw(new Line2D.Double(panelWidth/2,0, panelWidth/2 ,panelHeight));
        g2.setFont(new Font("Times New Roman", Font.PLAIN, 13));
 
        for(int i=1;i<getWidth();i+=9)
        {
            g2.draw(new Line2D.Double(i, (getHeight()/2)-2, i, (getHeight()/2)+2));
        }
 
        for(int i=1;i<getHeight();i+=9)
        {
            g2.draw(new Line2D.Double((getWidth()/2)-2, i,(getWidth()/2)+2,i));
        }
    }

	public void setValue(int iValue) {
		// dosomethingFancyWiththisValue(iValue);
		this.repaint();
		System.out.println("repainted");
	}
}

public class Graphs extends JFrame
{
    // renaming variables
    private GraphingPanel oGraphPanel = new GraphingPanel();
    private JPanel oPanel = new JPanel();
    private JLabel oLabel = new JLabel();
    private JTextField oField = new JTextField();
    private JButton oEnter = new JButton();
    private JButton oClear = new JButton();
    private JComboBox oSubjects = new JComboBox();
    private JComboBox oTopics = new JComboBox();
 
    public Graphs()
    {
        super();
        setSize(600, 500);
        setDefaultCloseOperation(EXIT_ON_CLOSE);
//        setUndecorated(true); 
        setLocationRelativeTo(null);
        loadUI();
    }
 
    public void loadUI()
    {
        oField = new JTextField();
        oPanel = new JPanel(null);
 

        oField.setBounds(240, 20, 120, 30);
        oField.addKeyListener(new KeyAdapter(){
        	 @Override
        	    public void keyPressed(KeyEvent e)
        	    {// always try to work with constant values - it's safer!
        	        if(e.getKeyCode()==KeyEvent.VK_ENTER) 
        	        {
        	        	String strValue = oField.getText();
        	        	System.out.println("Enter pressed. Value: " + strValue);
        	        	//You probably want numbers entered in the TextBox - so you should check for the entered Value to be a number:
        	        	validateIsNumber(strValue);
        	        	oGraphPanel.setValue(Integer.parseInt(strValue));
//        	            DrawGraph g = new DrawGraph(oField.getText());
        	        }
        	    }

        }
        		); 
 
        oPanel.add(oField);
        oPanel.add(oGraphPanel);
        add(oPanel);
        setVisible(true);
    }
    
    private void validateIsNumber(String strValue) {
    	try{
    		Integer.parseInt(strValue); // just trying to parse the number, exception pops when not possible
    	} catch(Exception oException){
    		 JOptionPane.showMessageDialog(this, oException.getLocalizedMessage(),
                    "oooops", JOptionPane.ERROR_MESSAGE);
    	}
	}
    
 
    public static void main(String[] args)
    {
        new Graphs();
    }
 
   
}