[从头学数学] 第151节二次函数小结与复习题

最新推荐文章于 2021-07-26 17:20:15 发布

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本文介绍了一个使用JavaScript实现的矩阵类，该类能够帮助解决二次函数的问题，并通过具体的例子展示了如何利用行列式求解未知数及绘制二次函数图像。

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了筑基后期的修炼，
这次要修炼的目标是[二次函数小结与复习题]。

正剧开始：

星历2016年03月24日 14:45:50, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[二次函数小结与复习题]。

话说最近[人叫板老师]老爱让[机器小伟]解方程，本来吧这也没什么，毕竟用python的numpy库可以轻松搞定。

但是呢，[工程师阿伟]却嫌那个numpy要加载的东西太多，也不知道里面都有些什么，反正附带的文件相当的多。

而且最重要的是，它是用python写的，所以阿伟就想用JS自己写一个，当然，这只能慢慢来。

考虑到[人叫板老师]现在的方程也就是一些二元、三元的，好像用行列式也可以解决了。

于是，阿伟终于忍不住了，还是走上了从头创建矩阵类的道路。

<span style="font-size:18px;">/**
* @usage   矩阵类
* @author  mw
* @date    2016年03月24日  星期四  14:51:45 
* @param
* @return
*
*/
function Matrix() {
	this.delta = function(matrix) {
		var size = matrix.length;
		
		if (!matrix[0].length) {
			return 0;
		}
		else {
			if (matrix[0].length != size) {
				return 0;
			}
			
			if (size < 1) {
				return 0;
			}
			else if (size == 1) {
				return matrix[0][0];
			}
			else {
				var d = 0;
				var subMatrix = [];
				
				for (var i = 0; i < size; i++) {
					subMatrix = this.subSet(matrix, i, 0);
					d += Math.pow(-1, i)*matrix[i][0]*this.delta(subMatrix);
				}
				return d;
			}
		}
	}
	
	this.subSet = function(matrix, row, col) {	
		var size = matrix.length;
		
		if (!matrix[0].length) {
			return [];
		}
		else {
			if (matrix[0].length != size) {
				return [];
			}
			
			var newMatrix = new Array();
			var rowArray= new Array();
			
			for (var i = 0; i < size; i++) {
				if (i == row) {
					continue;
				}
				else {
					rowArray = [];
				
					for (var j = 0; j < size; j++) {
						if (j == col) {
							continue;
						}
						else {
							rowArray.push(matrix[i][j]);
						}
					}
					
					newMatrix.push(rowArray)
				}
			}
			
			return newMatrix;
		}
	}
	
	this.identityMatrix = function(rank) {
		var newMatrix = new Array();
		var rowArray= new Array();
		
		for (var i = 0; i < rank; i++) {
			rowArray = [];
			
			for (var j = 0; j < rank; j++) {
				if (j == i) {
					rowArray.push(1);
				}
				else {
					rowArray.push(0);
				}
			}
			
			newMatrix.push(rowArray);
		}
		
		return newMatrix;
	}
	
	//把从1-n的n个数按顺序放到矩阵中
	this.sequenceMatrix = function(rank) {
		var newMatrix = new Array();
		var rowArray= new Array();
		
		var count = 0;
		for (var i = 0; i < rank; i++) {
			rowArray = [];
			
			for (var j = 0; j < rank; j++) {
				count++;
				rowArray.push(count);
				
			}
			
			newMatrix.push(rowArray);
		}
		
		return newMatrix;
	}
	
	
	this.print = function(matrix) {
		var size = matrix.length;
		
		if (!matrix[0].length) {
			return [];
		}
		else {
			var rSize = matrix[0].length;
			
			var s = '';
			
			for (var i = 0; i < size; i++) {
				s = '';
				
				for (var j = 0; j < rSize; j++) {
					s += matrix[i][j].toFixed(2) + ' , ';
				}
				
				document.write(s+'<br/>');
			}
		}
	}
					
					
	
}</span>

现在这么点函数能做什么呢？

<span style="font-size:18px;">	var matrix = new Matrix();
	var m = matrix.identityMatrix(5);
	var m2 = [[1,5,7],[2,6,-3],[99,23,5]];
	matrix.print(m);
	document.write(matrix.delta(m)+'<br/>');
		matrix.print(m2);
	document.write(matrix.delta(m2)+'<br/>');
	
	var m3 = [[4,5,7,3],[4,-2,6,-1],[5,8,5,3],[14,67,4,2]];
			matrix.print(m3);
	document.write(matrix.delta(m3)+'<br/>');</span>

还是来看看这次[人叫板老师]出的题吧。

也不知道能不能用上这个工具。

<span style="font-size:18px;">		var r = 20;  
        config.setSector(1,1,1,1);    
        config.graphPaper2D(0, 0, r);  
        config.axis2D(0, 0, 180, 1);    
      
        var scaleX = 2*r, scaleY = 2*r;
        var spaceX = 2, spaceY = 10; 
		var xS = -10, xE = 10;
		var yS = -100, yE = 100;
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');  
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');  
          
        var transform = new Transform();  
        var a = [], b = [], c = [], d = [];
		
		for (var x = xS; x <= xE; x+=spaceX/2) {
			a.push([x, x*x-2*x-3]);
			b.push([x, 1+6*x-x*x]);
			c.push([x, 0.5*x*x+2*x+1]);
			d.push([x, -1/4*x*x+x-4]);
		}
		
		
          
        a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY); 
		
          
        var tmp = [].concat(a);  
        shape.pointDraw(tmp, 'red');  
        tmp = [].concat(a);  
        shape.multiLineDraw(tmp, 'pink');
		
		plot.setFillStyle('red');
		plot.fillText('y = x*x-2*x-3', 100, -20, 200);
		
		
		
		b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);
		tmp = [].concat(b);  
        shape.pointDraw(tmp, 'blue');  
        tmp = [].concat(b);  
        shape.multiLineDraw(tmp, '#22CCFF');  
		plot.setFillStyle('blue');
		plot.fillText('y=1+6*x-x*x', 100, -40, 200);
		

          
		c = transform.scale(transform.translate(c, 0, 0), scaleX/spaceX, scaleY/spaceY);
        tmp = [].concat(c);  
        shape.pointDraw(tmp, 'green');  
        tmp = [].concat(c);  
        shape.multiLineDraw(tmp, '#CCFF22');
		plot.setFillStyle('green');
		plot.fillText('y=0.5*x*x+2*x+1', 100, -60, 200);
		
		d = transform.scale(transform.translate(d, 0, 0), scaleX/spaceX, scaleY/spaceY);
        tmp = [].concat(d);  
        shape.pointDraw(tmp, 'orange');  
        tmp = [].concat(d);  
        shape.multiLineDraw(tmp, '#CC8800');
		plot.setFillStyle('orange');
		plot.fillText('y=-1/4*x*x+x-4', 100, -80, 200);
		</span>

先用行列式解出a, b,c

<span style="font-size:18px;">	var matrix = new Matrix();
	var matrixArray = new Array();
	var rowArray = new Array();
	var ma, mb, mc;
	
	var a = [[-3,2],[-1,-1],[1,3]];
	
	matrixArray = [[9,-3,1],[1,-1,1],[1,1,1]];
	
	
	ma = matrix.deepCopy(matrixArray);
	for (var i = 0; i < 3; i++) {
		ma[i][0] = a[i][1];
	}
	mb = matrix.deepCopy(matrixArray);
	for (var i = 0; i < 3; i++) {
		mb[i][1] = a[i][1];
	}
	mc = matrix.deepCopy(matrixArray);
	for (var i = 0; i < 3; i++) {
		mc[i][2] = a[i][1];
	}
	
	
	var d, da, db, dc;
	d = matrix.delta(matrixArray);
	da = matrix.delta(ma);
	db = matrix.delta(mb);
	dc = matrix.delta(mc);	
	
	
	matrix.print(matrixArray);
	
	document.write(d+'<br/>');
		matrix.print(ma);
	document.write(da+'<br/>');
		matrix.print(mb);
	document.write(db+'<br/>');
		matrix.print(mc);
	document.write(dc+'<br/>');
	
	var s = 'a = '+ (da/d).toFixed(2)+', b= '+(db/d).toFixed(2)+', c = ' + (dc/d).toFixed(2);
	document.write(s+'<br/>');</span>

再画图

<span style="font-size:18px;">		var r = 20;  
        config.setSector(1,1,1,1);    
        config.graphPaper2D(0, 0, r);  
        config.axis2D(0, 0, 180, 1.5);    
      
        var scaleX = 2*r, scaleY = 2*r;
        var spaceX = 2, spaceY = 2; 
		var xS = -10, xE = 10;
		var yS = -10, yE = 20;
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');  
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');  
          
        var transform = new Transform();  
        var a = [], b = [], c = [], d = [];
		
		for (var x = xS; x <= xE; x+=spaceX/5) {
			a.push([x, 0.88*x*x+2*x+0.13]);
			
		}
		
		
          
        a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY); 
		
          
        var tmp = [].concat(a);  
        shape.pointDraw(tmp, 'red');  
        tmp = [].concat(a);  
        shape.multiLineDraw(tmp, 'pink');
		
		plot.setFillStyle('red');
		plot.fillText('y = 0.88*x*x+2*x+0.13', 100, -20, 200);</span>

<span style="font-size:18px;">	var matrix = new Matrix();
	var matrixArray = new Array();
	var rowArray = new Array();
	var ma, mb, mc;
	
	var a = [[-0.5,0],[1.5,0],[0,-5]];
	
	for (var i = 0; i < 3; i++) {
		matrixArray.push([a[i][0]*a[i][0], a[i][0], 1]);
	}
	
	
	ma = matrix.deepCopy(matrixArray);
	for (var i = 0; i < 3; i++) {
		ma[i][0] = a[i][1];
	}
	mb = matrix.deepCopy(matrixArray);
	for (var i = 0; i < 3; i++) {
		mb[i][1] = a[i][1];
	}
	mc = matrix.deepCopy(matrixArray);
	for (var i = 0; i < 3; i++) {
		mc[i][2] = a[i][1];
	}
	
	
	var d, da, db, dc;
	d = matrix.delta(matrixArray);
	da = matrix.delta(ma);
	db = matrix.delta(mb);
	dc = matrix.delta(mc);	
	
	
	matrix.print(matrixArray);
	
	document.write(d+'<br/>');
		matrix.print(ma);
	document.write(da+'<br/>');
		matrix.print(mb);
	document.write(db+'<br/>');
		matrix.print(mc);
	document.write(dc+'<br/>');
	
	var s = 'a = '+ (da/d).toFixed(2)+', b= '+(db/d).toFixed(2)+', c = ' + (dc/d).toFixed(2);
	document.write(s+'<br/>');
	

</span>

<span style="font-size:18px;">0.25 , -0.50 , 1.00 , 
2.25 , 1.50 , 1.00 , 
0.00 , 0.00 , 1.00 , 
1.5
0.00 , -0.50 , 1.00 , 
0.00 , 1.50 , 1.00 , 
-5.00 , 0.00 , 1.00 , 
10
0.25 , 0.00 , 1.00 , 
2.25 , 0.00 , 1.00 , 
0.00 , -5.00 , 1.00 , 
-10
0.25 , -0.50 , 0.00 , 
2.25 , 1.50 , 0.00 , 
0.00 , 0.00 , -5.00 , 
-7.5
a = 6.67, b= -6.67, c = -5.00</span>

<span style="font-size:18px;">		var r = 20;  
        config.setSector(1,1,1,1);    
        config.graphPaper2D(0, 0, r);  
        config.axis2D(0, 0, 180, 1.5);    
      
        var scaleX = 2*r, scaleY = 2*r;
        var spaceX = 2, spaceY = 2; 
		var xS = -10, xE = 10;
		var yS = -10, yE = 20;
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');  
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');  
          
        var transform = new Transform();  
        var a = [], b = [], c = [], d = [];
		
		for (var x = xS; x <= xE; x+=spaceX/5) {
			a.push([x, 6.67*x*x-6.67*x-5]);
			
		}
		
		
          
        a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY); 
		
          
        var tmp = [].concat(a);  
        shape.pointDraw(tmp, 'red');  
        tmp = [].concat(a);  
        shape.multiLineDraw(tmp, 'pink');
		
		plot.setFillStyle('red');
		plot.fillText('y = 6.67*x*x-6.67*x-5', 100, -20, 200);</span>

<span style="font-size:18px;">		var r = 20;  
        config.setSector(1,1,1,1);    
        config.graphPaper2D(0, 0, r);  
        config.axis2D(0, 0, 180, 1.5);    
      
        var scaleX = 2*r, scaleY = 2*r;
        var spaceX = 5, spaceY = 150; 
		var xS = -10, xE = 20;
		var yS = -10, yE = 800;
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');  
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');  
          
        var transform = new Transform();  
        var a = [], b = [], c = [], d = [];
		
		for (var x = xS; x <= xE; x+=spaceX/5) {
			a.push([x, 6.28*x*(18-x)]);
			
		}
		
		
          
        a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY); 
		
          
        var tmp = [].concat(a);  
        shape.pointDraw(tmp, 'red');  
        tmp = [].concat(a);  
        shape.multiLineDraw(tmp, 'pink');
		
		plot.setFillStyle('red');
		plot.fillText('y = 6.28*x*(18-x)', 100, -20, 200);</span>