Month: April 2012

The Algebra of Straight Lines

A quick graphical calculator for straight lines — based on This. It’s still incomplete, but it’s functional, and a useful complement to the equation of a straight line animation and the parabola graphing.

You can graph lines based on the equation of a line (in either the slope-intercept or point-slope forms), or from two points.

[inline]

Straight Lines


Slope-Intercept Form Two Points Point-Slope Form

y = m x + b
(x1, y1) =
( ,
)
y – y1 = m ( x – x1)
y =
x +
(x2, y2) =
( ,
) 		y –
=
( x –
)
x intercept:

Slope:

Equation:
y intercept:

Points:


Your browser does not support the canvas element.

[script type=”text/javascript”]

var width=500;
var height=500;
var xrange=10;
var yrange=10;

mx = width/(2.0*xrange);
bx = width/2.0;
my = -height/(2.0*yrange);
by = height/2.0;

document.getElementById(‘testing’).innerHTML = “Hello2”;

function draw_9359(ctx, polys) {

t_9359=t_9359+dt_9359;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
ctx.lineWidth=2;
polys[0].draw(ctx);

polys[0].write_eqn2(ctx);

//ctx.fillText (polys[0].get_parabola_vertex_form_eqn(), xp(5), yp(-5) );

// SHOW 2 POINTS USED TO CALCULATE LINE
if (show_2pts_ctrl_9359.checked==true) {
polys[0].circle_point(ctx, polys[0].p1.x, polys[0].p1.y, “#000”, 0.15);
polys[0].label_point_on_line(ctx, polys[0].p1.x, polys[0].p1.y);
polys[0].circle_point(ctx, polys[0].p2.x, polys[0].p2.y, “#000”, 0.15);
polys[0].label_point_on_line(ctx, polys[0].p2.x, polys[0].p2.y);

}

// SHOW Y INTERCEPT
if (show_y_intercept_ctrl_9359.checked==true) {
polys[0].draw_y_intercept_line(ctx);
document.getElementById(‘y_intercept_pos_9359’).innerHTML = “( 0 , “+ polys_9359[0].c.toPrecision(2)+ ” )”;

if (polys[0].b > 0.0) {
ctx.textAlign=”right”;
xoff = -0.5;
}
else {
ctx.textAlign=”left”;
xoff = 0.5;
}
ctx.fillText (‘y intercept: y = ‘+polys[0].c.toPrecision(2), xp(xoff), yp(polys_9359[0].c+0.25));

} else { document.getElementById(‘y_intercept_pos_9359’).innerHTML = “”;}

// SHOW SLOPE
if (show_slope_ctrl_9359.checked==true) {
polys[0].draw_slope_line(ctx);
document.getElementById(‘slope_pos_9359’).innerHTML = ” = “+polys[0].b.toPrecision(2);

} else { document.getElementById(‘slope_pos_9359’).innerHTML = “”;}

document.getElementById(‘slope_int_equation_9359’).innerHTML = polys[0].get_eqn2();

//SHOW INTERCEPTS
ctx.textAlign=”center”;
if (show_intercepts_ctrl_9359.checked==true) {
polys[0].x_intercepts(ctx);
if (polys[0].order == 2) {
if (polys[0].x_intcpts.length > 0) {
line = “0 = “;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
line = line + “(x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ “)”;
}
ctx.fillText (line, xp(6), yp(7));
for (var i=0; i 0) {
if (polys[0].b > 0.0) {
ctx.textAlign=”right”;
xoff = -0.5;
}
else {
ctx.textAlign=”left”;
xoff = 0.5;
}
ctx.font = “12pt Calibri”;
ctx.fillStyle = “#00f”;
ctx.fillText (‘x intercept: x = ‘+polys[0].x_intcpts[0].toPrecision(2), xp(polys[0].x_intcpts[0]+xoff), yp(0.5));
document.getElementById(‘intercepts_pos_9359’).innerHTML = “( “+ polys[0].x_intcpts[0].toPrecision(2) + “, 0 )”;
}
else {
ctx.fillText (‘None’, xp(7), yp(7));
}

}

}
else {
document.getElementById(‘intercepts_pos_9359’).innerHTML = ” “;
}

// Finding the slope
outln = “

Finding the Slope

“;
outln += “

The slope of the line between the two points is the rise divided by the run: the change in elevation divided by the horizontal distance between the points.”;

// Write out intercept solution
outln = “

Finding the x-intercept

“;
outln += “

Start with the slope-intercept form of the equation:”;
outln += “

     “+polys[0].get_eqn2(“y”,”x”,”html”)+”“;
outln += “

Know that at the x-intercept, y = 0 , which you substitute into the equation to get:”;
outln += “

     “+polys[0].get_eqn2(“0″,”x”,”html”)+”“;
outln += “

Now move the b coefficient to the other side of the equation:”;
outln += “

     “+(-polys[0].c).toPrecision(2)+” = “+ polys[0].b.toPrecision(2)+”x”+”“;
outln += “

And divide both sides by the slope to solve for x:”;
outln += “

     “+(-polys[0].c).toPrecision(2)+ ” / “+ polys[0].b.toPrecision(2) + ” = “+ polys[0].b.toPrecision(2)+”x”+ ” / “+ polys[0].b.toPrecision(2)+”“;
outln += “

     “+(-polys[0].c/polys[0].b).toPrecision(2)+” = “+”x”+”“;
outln += “

Therefore, the line intercepts the x axis at the point:”;
outln += “

     ( “+(-polys[0].c/polys[0].b).toPrecision(2)+” , 0 )”;
outln += “

When,”;
outln += “

     “+”x = “+ (-polys[0].c/polys[0].b).toPrecision(2) +”“;

document.getElementById(‘x_intercept_9359’).innerHTML = outln;

//Write out the solution by factoring
solution = “

Finding the y-intercept

“;
if (polys[0].order == 2) {
solution = solution + “y = “+polys[0].a.toPrecision(2)+” x2 “+polys[0].bsign+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;

solution = solution + ‘Factoring:      ‘;

if (polys[0].x_intcpts.length > 0) {
solution = solution + ‘0 = ‘;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
solution = solution + “(x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ “)”;
}
solution = solution + ‘

‘;
solution = solution + ‘Set each factor equal to zero:
     ‘;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
solution = solution + “x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ ” = 0           “;
}
solution = solution + ‘

and solve for x:
     ‘;
for (var i=0; i‘;
}
document.getElementById(‘equation_9359’).innerHTML = solution;
}

else if (polys[0].order == 1) {
solution = solution + ‘

At the y-axis x = 0.
To find the y intercept, take the equation of the line, set x = 0, and solve for x.’;
solution = solution + ‘

     ‘;
solution = solution + “y = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;
solution = solution + ‘     ‘;
solution = solution + ” 0 = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;
solution = solution + ‘     ‘;
solution = solution + (-1.0*polys[0].c).toPrecision(2) +” = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+”

“;
solution = solution + ‘     ‘;
solution = solution + (-1.0*polys[0].c).toPrecision(2)+”/”+polys[0].b.toPrecision(2)+” = “+” x “+”

“;
solution = solution + ‘     ‘;
solution = solution + “x = “+ (-1.0*polys[0].c/polys[0].b).toPrecision(4)+”

“;

document.getElementById(‘equation_9359’).innerHTML = solution;

}

}

function update_form_9359 () {
b_coeff_9359.value = polys_9359[0].b+””;
c_coeff_9359.value = polys_9359[0].c+””;

y1_coeff_9359.value = polys_9359[0].y1+””;
b1_coeff_9359.value = polys_9359[0].b+””;
x1_coeff_9359.value = polys_9359[0].x1+””;

p1_x_9359.value = polys_9359[0].p1.x+””;
p1_y_9359.value = polys_9359[0].p1.y+””;
p2_x_9359.value = polys_9359[0].p2.x+””;
p2_y_9359.value = polys_9359[0].p2.y+””;
}

//init_mouse();

var c_9359=document.getElementById(“myCanvas_9359”);
var ctx_9359=c_9359.getContext(“2d”);

var change = 0.0001;

function create_lines_9359 () {
//draw line
//document.write(“hello world! “);
var polys = [];
polys.push(addPoly(0,2, 3));
document.getElementById(‘testing’).innerHTML = “Hello”;
polys[0].set_2_points_line();
document.getElementById(‘testing’).innerHTML = “Hello4″;

// polys[1].color = ‘#8C8’;

return polys;
}

var polys_9359 = create_lines_9359();

var x1=xp(-10);
var y1=yp(1);
var x2=xp(10);
var y2=yp(1);
var dc_9359=0.05;

var t_9359 = 0;
var dt_9359 = 100;
//end_ct = 0;
var st_pt_x_9359 = 2;
var st_pt_y_9359 = 1;
var show_y_intercept_9359 = 1; //1 to show y intercept on startup
var show_intercepts_9359 = 0; // 1 to show the intercepts
var show_slope_9359 = 0; // 1 to show the slope
var show_2pts_9359 = 0; // 1 to show the two points from which line can be calculated

var move_dir_9359 = 1.0; // 1 for up

//document.getElementById(‘comment_spot’).innerHTML = polys_9359[0].a+” “+polys_9359[0].b+” “+polys_9359[0].c+” : “+polys_9359[0].h+” “+polys_9359[0].k+” “;

document.getElementById(‘testing’).innerHTML = “Hello3”;

//slope-intercept form
var b_coeff_9359 = document.getElementById(“b_coeff_9359”);
var c_coeff_9359 = document.getElementById(“c_coeff_9359”);

//point-slope form
var y1_coeff_9359 = document.getElementById(“y1_coeff_9359”);
var b1_coeff_9359 = document.getElementById(“b1_coeff_9359”);
var x1_coeff_9359 = document.getElementById(“x1_coeff_9359”);

//equation from two points
var p1_x_9359 = document.getElementById(“x1_9359”);
var p1_y_9359 = document.getElementById(“y1_9359”);
var p2_x_9359 = document.getElementById(“x2_9359”);
var p2_y_9359 = document.getElementById(“y2_9359”);

//options
var show_y_intercept_ctrl_9359 = document.getElementById(“show_y_intercept_9359”);
if (show_y_intercept_9359 == 0) {show_y_intercept_ctrl_9359.checked=false;
} else {show_y_intercept_ctrl_9359.checked=true;}

var show_intercepts_ctrl_9359 = document.getElementById(“show_intercepts_9359”);
if (show_intercepts_9359 == 0) {show_intercepts_ctrl_9359.checked=false;
} else {show_intercepts_ctrl_9359.checked=true;}

var show_slope_ctrl_9359 = document.getElementById(“show_slope_9359”);
if (show_slope_9359 == 0) {show_slope_ctrl_9359.checked=false;
} else {show_slope_ctrl_9359.checked=true;}

var show_2pts_ctrl_9359 = document.getElementById(“show_pts_9359”);
if (show_2pts_9359 == 0) {show_2pts_ctrl_9359.checked=false;
} else {show_2pts_ctrl_9359.checked=true;}

update_form_9359();

//document.write(“test= “+c_coeff_9359.value+” “+polys_9359[0].c);
setInterval(“draw_9359(ctx_9359, polys_9359)”, dt_9359);

b_coeff_9359.onchange = function() {
polys_9359[0].set_b(parseFloat(this.value));
polys_9359[0].update_point_slope_form_line();
polys_9359[0].set_2_points_line();
update_form_9359();
}
c_coeff_9359.onchange = function() {
//polys_9359[0].c = parseFloat(this.value);
polys_9359[0].set_c(parseFloat(this.value));
polys_9359[0].update_point_slope_form_line();
polys_9359[0].set_2_points_line();
update_form_9359();
}

y1_coeff_9359.onchange = function() {
//polys_9359[0].a = parseFloat(this.value);
polys_9359[0].y1 = parseFloat(this.value);
polys_9359[0].update_slope_intercept_form_line();
polys_9359[0].set_2_points_line();
update_form_9359();
}

b1_coeff_9359.onchange = function() {
polys_9359[0].set_b(parseFloat(this.value));
polys_9359[0].update_slope_intercept_form_line();
polys_9359[0].set_2_points_line();
update_form_9359();
}

x1_coeff_9359.onchange = function() {
polys_9359[0].x1 = parseFloat(this.value);
polys_9359[0].update_slope_intercept_form_line();
polys_9359[0].set_2_points_line();
polys_9359[0].set_order()
update_form_9359();
}

p1_x_9359.onchange = function() {
polys_9359[0].p1.x = parseFloat(this.value);
polys_9359[0].update_line_from_2pts();
polys_9359[0].set_order()
update_form_9359();
}

p2_x_9359.onchange = function() {
polys_9359[0].p2.x = parseFloat(this.value);
polys_9359[0].update_line_from_2pts();
polys_9359[0].set_order()
update_form_9359();
}

p1_y_9359.onchange = function() {
polys_9359[0].p1.y = parseFloat(this.value);
polys_9359[0].update_line_from_2pts();
polys_9359[0].set_order()
update_form_9359();
}

p2_y_9359.onchange = function() {
polys_9359[0].p2.y = parseFloat(this.value);
polys_9359[0].update_line_from_2pts();
polys_9359[0].set_order()
update_form_9359();
}

//draw_9359();
//document.write(“x”+x2+”x”);
//ctx_9359.fillText (“n=”, xp(5), yp(5));

[/script]

[/inline]

The Search for a New Earth

This NASA video updates us on the search for Earth-like planets around other stars. It overviews what’s been found, and outlines some upcoming missions.

The key to finding a planet hospitable to life (as we know it) is finding one with water at the surface. We’ve found large waterworlds that are too large and hot, with “thick, steamy atmosphere[s]”.

We’ve also found Earth-sized planets but they’re, mostly, too close to their stars to support liquid water, and it’s hard to tell what their atmospheres are like because they’re so far away. One of NASA’s upcoming missions, one will look at the light reflected off Earth-sized planets to determine the composition of atmospheres: the technique is called transit spectroscopy, and is based on detecting the emission spectra of the gasses in the atmosphere.

ᔥ Science@NASA ↬ Pale Blue Blog

How to Write a Research Paper

An excellent series by the American Chemical Society starts with the basics of, “How to Write a Paper to Communicate Your Research,” but also addresses the question of why publish your research. It ought to help my students understand why I’m so insistent on lab reports.

ᔥ ACS Publishing ↬ Pale Blue Blog