Everything You (N)ever Wanted to Know About Parabolas

So that my students could more easily check their answers graphically, I put together a page with a more complete analysis of parabolas (click this link for more details).

[inline]

Analyzing Parabolas

Standard Form	Vertex Form
`y = a x² + b x + c`	`y = a (x – h)² + k`
y = x² + x +	y = ( x – ) ² +

Intercepts:	Vertex:
Focus:	Directrix:	Axis:

Solution by Factoring:

y = x² x

[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;

function draw_9239(ctx, polys) {
t_9239=t_9239+dt_9239;
//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);

//polys[0].y_intercepts(ctx);
//write intercepts on graph

// SHOW VERTEX
if (show_vertex_ctrl_9239.checked==true) {
polys[0].draw_vertex(ctx);
document.getElementById(‘vertex_pos_9239’).innerHTML = “(“+polys_9239[0].vertex.x.toPrecision(2)+ ” , “+polys_9239[0].vertex.y.toPrecision(2)+ ” )”;
} else { document.getElementById(‘vertex_pos_9239’).innerHTML = “”;}

// SHOW FOCUS
if (show_focus_ctrl_9239.checked==true) {
polys[0].draw_focus(ctx);
document.getElementById(‘focus_pos_9239’).innerHTML = “(“+polys_9239[0].focus.x.toPrecision(2)+ ” , “+polys_9239[0].focus.y.toPrecision(2)+ ” )”;
} else { document.getElementById(‘focus_pos_9239’).innerHTML = “”;}

// SHOW DIRECTRIX
if (show_directrix_ctrl_9239.checked==true) {
polys[0].draw_directrix(ctx);
document.getElementById(‘directrix_pos_9239’).innerHTML = polys[0].directrix.get_eqn2(“y”,”x”,”html”);

polys[0].directrix.write_eqn2(ctx, polys[0].directrix.get_eqn2(“directrix: y”));
} else { document.getElementById(‘directrix_pos_9239’).innerHTML = “”;}

// SHOW AXIS
if (show_axis_ctrl_9239.checked==true) {
polys[0].draw_parabola_axis(ctx);
document.getElementById(‘axis_pos_9239’).innerHTML = “x = “+polys[0].vertex.x.toPrecision(2);
}

//SHOW INTERCEPTS
ctx.textAlign=”center”;
if (show_intercepts_ctrl_9239.checked==true) {
polys[0].x_intercepts(ctx);
ctx.fillText (‘x intercepts: (when y=0)’, xp(6), yp(8));
//ctx.fillText (‘intercepts=’+polys[0].x_intcpts.length, xp(5), yp(-5));
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) {
for (var i=0; i2 “+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_9239’).innerHTML = solution;
}

else if (polys[0].order == 1) {
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 + ‘

Set y=0 and solve for x:
‘;
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_9239’).innerHTML = solution;
}

}

function update_form_9239 () {
a_coeff_9239.value = polys_9239[0].a+””;
b_coeff_9239.value = polys_9239[0].b+””;
c_coeff_9239.value = polys_9239[0].c+””;

av_coeff_9239.value = polys_9239[0].a+””;
hv_coeff_9239.value = polys_9239[0].h+””;
kv_coeff_9239.value = polys_9239[0].k+””;

}

//init_mouse();

var c_9239=document.getElementById(“myCanvas_9239”);
var ctx_9239=c_9239.getContext(“2d”);

var change = 0.0001;

function create_lines_9239 () {
//draw line
//document.write(“hello world! “);
var polys = [];
polys.push(addPoly(1,6, 5));

// polys.push(addPoly(0.25, 1, 0));
// polys[1].color = ‘#8C8’;

return polys;
}

var polys_9239 = create_lines_9239();

var x1=xp(-10);
var y1=yp(1);
var x2=xp(10);
var y2=yp(1);
var dc_9239=0.05;

var t_9239 = 0;
var dt_9239 = 100;
//end_ct = 0;
var st_pt_x_9239 = 2;
var st_pt_y_9239 = 1;
var show_vertex_9239 = 1; //1 to show vertex on startup
var show_focus_9239 = 1; // 1 to show the focus
var show_intercepts_9239 = 1; // 1 to show the intercepts
var show_directrix_9239 = 1; // 1 to show the directrix
var show_axis_9239 = 1; //1 to show the axis of the parabola

var move_dir_9239 = 1.0; // 1 for up

//document.getElementById(‘comment_spot’).innerHTML = polys_9239[0].a+” “+polys_9239[0].b+” “+polys_9239[0].c+” : “+polys_9239[0].h+” “+polys_9239[0].k+” “;

//standard form
var a_coeff_9239 = document.getElementById(“a_coeff_9239”);
var b_coeff_9239 = document.getElementById(“b_coeff_9239”);
var c_coeff_9239 = document.getElementById(“c_coeff_9239”);

//vertex form
var av_coeff_9239 = document.getElementById(“av_coeff_9239”);
var hv_coeff_9239 = document.getElementById(“hv_coeff_9239”);
var kv_coeff_9239 = document.getElementById(“kv_coeff_9239”);

//options
var show_vertex_ctrl_9239 = document.getElementById(“show_vertex_9239”);
if (show_vertex_9239 == 0) {show_vertex_ctrl_9239.checked=false;
} else {show_vertex_ctrl_9239.checked=true;}

var show_focus_ctrl_9239 = document.getElementById(“show_focus_9239”);
if (show_focus_9239 == 0) {show_focus_ctrl_9239.checked=false;
} else {show_focus_ctrl_9239.checked=true;}

var show_intercepts_ctrl_9239 = document.getElementById(“show_intercepts_9239”);
if (show_intercepts_9239 == 0) {show_intercepts_ctrl_9239.checked=false;
} else {show_intercepts_ctrl_9239.checked=true;}

var show_directrix_ctrl_9239 = document.getElementById(“show_directrix_9239”);
if (show_directrix_9239 == 0) {show_directrix_ctrl_9239.checked=false;
} else {show_directrix_ctrl_9239.checked=true;}

var show_axis_ctrl_9239 = document.getElementById(“show_axis_9239”);
if (show_axis_9239 == 0) {show_axis_ctrl_9239.checked=false;
} else {show_axis_ctrl_9239.checked=true;}

update_form_9239();

//document.write(“test= “+c_coeff_9239.value+” “+polys_9239[0].c);
setInterval(“draw_9239(ctx_9239, polys_9239)”, dt_9239);

a_coeff_9239.onchange = function() {
//polys_9239[0].a = parseFloat(this.value);
polys_9239[0].set_a(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}
b_coeff_9239.onchange = function() {
//polys_9239[0].b = parseFloat(this.value);
polys_9239[0].set_b(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}
c_coeff_9239.onchange = function() {
//polys_9239[0].c = parseFloat(this.value);
polys_9239[0].set_c(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}

av_coeff_9239.onchange = function() {
//polys_9239[0].a = parseFloat(this.value);
polys_9239[0].set_a(parseFloat(this.value));
polys_9239[0].update_standard_form_parabola();
update_form_9239();
}

hv_coeff_9239.onchange = function() {
polys_9239[0].h = parseFloat(this.value);
polys_9239[0].update_standard_form_parabola();
polys_9239[0].set_order()
update_form_9239();
}

kv_coeff_9239.onchange = function() {
polys_9239[0].k = parseFloat(this.value);
polys_9239[0].update_standard_form_parabola();
polys_9239[0].set_order()
update_form_9239();
}

//draw_9239();
//document.write(“x”+x2+”x”);
//ctx_9239.fillText (“n=”, xp(5), yp(5));

[/script]

[/inline]

Converting the forms

The key relationships are the ones to convert from the standard form of the parabolic equation:

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

(1)

to the vertex form:

$y = a (x - h)^2 + k$

(2)

If you multiply out the vertex equation form you get:

y = a x² – 2ah x + ah² + k

(3)

When you compare this equation to the standard form of the equation (Equation 1), if you look at the coefficients and the constants, you can see that:

To convert from the vertex to the standard form use:

         $a = a$ (4)

         $b = -2ah$ (5)

         $c = ah^2 + k$ (6)

Going the other way,

To convert from the standard to the vertex form of parabolic equations use:

         $a = a$
(7)

         $h = \frac{-b}{2a}$
(8)

         $k = c - ah^2$
(9)

Although it is sometimes convenient to let k not depend on coefficients from its own equation:

$k = c - \frac{b^2}{4a}$ (10)

The Vertex and the Axis

The nice thing about the vertex form of the equation of the parabola is that if you want the find the coordinates of the vertex of the parabola, they’re right there in the equation.

Specifically, the vertex is located at the point:

$(x_v, y_v) = (h, k)$

(11)

The axis of the parabola is the vertical line going through the vertex, so:

The equation for the axis of a parabola is:

$x = h$

(12)

Focus and Directrix

Finally, it’s important to note that the distance (d) from the vertex of the parabola to its focus is given by:

$d = \frac{1}{4a}$

(13)

Which you can just add d on to the coordinates of the vertex (Equation 11) to get the location of the focus.

$(x_f, y_f) = (x_v, y_v + d)$

(14)

The directrix is just the opposite, vertical distance away, so the equation for the directrix is the equation of the horizontal line at:

$y = y_v + d$

(15)

References

There are already some excellent parabola references out there including:

Wonderful interactive graphs by Murray Bourne.
Nice simple explanations of the different forms of the equations.
More complex explanations from Wolfram Alpha.

Enhancing Creativity Just by Doing Things a Little Bit Differently

When my verbal students complain about having to do diagrams (or vice versa) I try to explain that it’s useful for them to see, and be able to learn from, different points of view. There also a body of research showing that breaking familiar routines enhances creativity.

Actively doing something different, just by changing your routine for example (milk first then cereal instead of the other way around), seems to improve people’s cognitive flexibility (see: Ritter et al., 2011), but you have to do it yourself.

… being exposed to simple unconventional events, such as preparing breakfast in the “wrong” order, increased cognitive flexibility. Furthermore, these effects were found only when people actively participated in the unconventional activities. Just seeing someone else perform the activities was not enough (my emphasis).

— Damian, R., 2012: Why Would Doing Something Unconventionally Make Us More Creative? in Science + Religion Today.

These findings also suggest that greater diversity also promotes creativity — periods of greater immigration have been followed by increases in innovation.

Challenge vs. skill, showing "flow" region. (Image and caption by Wikipedia User:Oliverbeatson).

It also suggest one reason why a little challenge is essential for motivating students to get into the flow zone for learning. People learn more, and become more engaged, when they’re challenged, have to struggle a little, and think differently, while figuring things out for themselves.

An interesting interview with Csikszentmihalyi is here.

How to Cite a Tweet

Last Name, First Name. (User Name). “Entire text of tweet (don’t change the CapitalizatioN)”, Date, Time. Tweet.

It’s essential to give credit where it’s due, especially in academic papers, but it’s good practice anywhere, anytime, and with anything. Thus the MLA has come up with a standard format for citing Tweets.

Begin the entry in the works-cited list with the author’s real name and, in parentheses, user name, if both are known and they differ. If only the user name is known, give it alone.

Next provide the entire text of the tweet in quotation marks, without changing the capitalization. Conclude the entry with the date and time of the message and the medium of publication (Tweet). For example:

Athar, Sohaib (ReallyVirtual). “Helicopter hovering above Abbottabad at 1AM (is a rare event).” 1 May 2011, 3:58 p.m. Tweet.

— Modern Language Association (MLA), 2012: How do I cite a tweet?

(via Madrigal, 2012: How Do You Cite a Tweet in an Academic Paper?)

Want to See Your DNA?

NOVA shows how you can see your own DNA at home. (Learn.Genetics has more details about why it works.)

Euler’s Method for Approximating a Solution to a Differential Equation

Based on Euler’s Method, this interactive graph illustrates a numerical method for solving differential equations. This approach is at the core of many sophisticated computer models of physical phenomena (like climate and weather).

[inline]

Starting point: (x,y) = ( , )
Step size:
Direction:

Slope equation: dy/dx = x +
Show analytical solution:

If you know the equation for the slope of a curve (the red line for example),

and a point that the curve passes through, such as , you can integrate to find the equation of the curve:

[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;

function upause () {
ct = 1;
}

function draw_9110(ctx, polys) {
t_9110=t_9110+dt_9110;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
polys[0].slopeField(ctx, 1.0, 1.0, 0.5);
ctx.lineWidth=2;
polys[0].draw(ctx);
polys[0].write_eqn(ctx, “slope = dy/dx”);

eu = eu_9110;
//override nsteps values
//max_nsteps_9110 = eu.nsteps;
//eu.nsteps = nsteps_9110;

if (nsteps_9110 <= max_nsteps_9110) { nsteps_9110 = nsteps_9110 + 1;} else { nsteps_9110 = 0; } eu.nsteps = nsteps_9110; if (show_parabola_9110 == 1) { polys[0].Integrate(ctx, eu.x1, eu.y1); polys[0].integral.color = '#3c3'; polys[0].integral.draw(ctx); } pos = polys[0].EulerApprox(ctx, eu); //document.getElementById('comment_9110').innerHTML = eu.nsteps+ " " + eu.x1 + " " + eu.y1+ " " + eu.dx+ " " + eu.dir+ " " + max_nsteps_9110; //starting point label ctx.textAlign="right"; ctx.textBaseline="alphabetic"; ctx.font = "14pt Calibri"; ctx.fillStyle = "#00f"; ctx.fillText ('start = ('+eu.x1+','+eu.y1+")", xp(eu.x1-0.25), yp(eu.y1-0.5)); //pos = polys[0].EulerApprox(ctx, st_pt_x_9110, st_pt_y_9110, 0.5,20,-1); //pos = polys[0].EulerApprox(ctx, st_pt_x_9110, st_pt_y_9110, 1, 6,1); //draw starting point ctx.strokeStyle="#00f"; ctx.lineWidth=2; ctx.beginPath(); ctx.arc(xp(eu.x1),yp(eu.y1),dxp(0.1),0,Math.PI*2); ctx.closePath(); ctx.stroke(); //ctx.fillText (' c='+move_dir_9110+' '+polys[1].c, xp(5), yp(5)); } //init_mouse(); var c_9110=document.getElementById("myCanvas_9110"); var ctx_9110=c_9110.getContext("2d"); var change = 0.0001; function create_lines_9110 () { //draw line var polys = []; polys.push(addPoly(0, 0.5, -1)); return polys; } function set_max_nsteps_9110 (eu) { max_nsteps_9110 = 1.5*(xrange - eu.x1)/eu.dx; } var polys_9110 = create_lines_9110(); var x1=xp(-10); var y1=yp(1); var x2=xp(10); var y2=yp(1); var dc_9110=0.05; var t_9110 = 0; var dt_9110 = 500; var st_dx_9110 = 1.0; var st_nsteps_9110 = 0; var nsteps_9110 = 0; var max_nsteps_9110 = 20; var dir_9110 = 1; var show_parabola_9110 = 1; var st_pt_x_9110 = -2; var st_pt_y_9110 = -3; var move_dir_9110 = 1.0; // 1 for up //Form interaction function update_form_9110 (x, y, dx, nsteps, dir, b, c, show_para) { x_9110.value = st_pt_x_9110+""; y_9110.value = st_pt_y_9110+""; dx_9110.value = st_dx_9110.toPrecision(2); //c_max_nsteps_9110.value = max_nsteps_9110+""; //document.getElementById('comment_9110').innerHTML = c_dir_9110; if (dir == 1) { c_dir_9110.selectedIndex = 0; } else {c_dir_9110.selectedIndex = 1;} b_9110.value = b+""; c_9110.value = c+""; if (show_para == 1) { ctrl_show.checked = 1; } else {ctrl_show.checked = 0;} } function get_euler_form_9110 () { pos = {x : parseFloat(x_9110.value), y : parseFloat(y_9110.value)} if (c_dir_9110.selectedIndex == 0) { dir = 1;} else {dir = -1;} eu = addEuler(pos.x, pos.y, parseFloat(dx_9110.value), max_nsteps_9110, dir); return eu } function update_solution_9110 (ctx, eu) { //update equations and solution document.getElementById('slope_eqn_9110').innerHTML = polys_9110[0].get_eqn("dy/dx"); //document.getElementById('eqn_9110').innerHTML = "hey"+eu.y1; polys_9110[0].Integrate(ctx, eu.x1, eu.y1); document.getElementById('eqn_9110').innerHTML = polys_9110[0].integral.get_eqn(); document.getElementById('init_pt_9110').innerHTML = "("+eu.x1+","+eu.y1+")"; } function add_euler_line() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } var x_9110 = document.getElementById("x_9110"); var y_9110 = document.getElementById("y_9110"); var dx_9110 = document.getElementById("dx_9110"); //var c_max_nsteps_9110 = document.getElementById("nsteps_9110"); var c_dir_9110 = document.getElementById("dir_9110"); var b_9110 = document.getElementById("b_9110"); var c_9110 = document.getElementById("c_9110"); var ctrl_show = document.getElementById("chk_show_para_9110"); update_form_9110(x_9110, y_9110, dx_9110, nsteps_9110, dir_9110, polys_9110[0].b, polys_9110[0].c, show_parabola_9110); var eu_9110 = get_euler_form_9110 (); set_max_nsteps_9110(eu_9110); update_solution_9110(ctx_9110, eu_9110); setInterval("draw_9110(ctx_9110, polys_9110)", dt_9110); x_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } y_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } c_dir_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } dx_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; //max_nsteps_9110 = 1.4 * (10 - eu_9110.x1)/eu_9110.dx; set_max_nsteps_9110(eu_9110); nsteps_9110 = 0; } b_9110.onchange = function() { polys_9110[0].set_b(parseFloat(this.value)); eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } c_9110.onchange = function() { polys_9110[0].set_c(parseFloat(this.value)); eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } ctrl_show.onchange = function() { if (this.checked == 1) { show_parabola_9110 = 1;} else {show_parabola_9110 = 0} } [/script] [/inline]

If you don’t have a starting point (initial condition), you can draw a slope field to see what the general pattern of all the possible solutions.

Even with a starting point, however, there are just times when you can’t integrate the slope equation — it’s either too difficult or even impossible.

Then, what you can do is come up with an approximation of what the curve looks like by projecting along the slope from the starting point.

The program above demonstrates how it’s done. This approach is called Euler’s Method, and is gives a numerical approximation rather than finding the exact, analytical solution using calculus (integration).

So why use an approximation when you can find the exact solution? Because, there are quite a number of problems that are impossible or extremely difficult to solve analytically, things like: the diffusion of pollution in a lake; how changing temperature in the atmosphere gives you weather and climate; the flow of groundwater in aquifers; stresses on structural members of buildings; and the list goes on and on.

As with most types of numerical approximations, you get better results if you can reduce the step size between projections of the slope. Try changing the numbers and see.

A more detailed version, with solutions, is here: Euler’s Method.

A good reference: Euler’s Method by Paul Dawkins.

Taylor Mali: Humorous Anecdotes from a Teacher

(hat tip: Mrs. D. our head of school).

Solving Quadratic Equations

So here we have a little grapher that solves quadratic equations visually. Just enter the coefficients in the equation.

[inline]

Quadratic Equation: y = a x² + b x + c

Enter:
y =
x² +
x +

Solution:

Analytical solution by factoring (with a little help from the quadratic equation if necessary).

[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;

function draw_9114(ctx, polys) {
t_9114=t_9114+dt_9114;
//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_eqn(ctx);

//polys[0].y_intercepts(ctx);
polys[0].x_intercepts(ctx);
//write intercepts on graph

ctx.fillText (‘x intercepts: (when y=0)’, xp(9), yp(8));
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(9), yp(7));
// if (polys[0].order == 2 ) {
// if (polys[0].x_intcpts[0] > 0.0) { sign1=”-“;} else {sign1=”+”;}
// if (polys[0].x_intcpts[1] > 0.0) { sign2=”-“;} else {sign2=”+”;}
// ctx.fillText (‘0 = (x ‘+sign1+” “+Math.abs(polys[0].x_intcpts[0])+”) (x “+sign2+Math.abs(polys[0].x_intcpts[1])+”)”, xp(9), yp(7));
// }
for (var i=0; i2 “+polys[0].bsign+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;

solution = solution + ‘Factoring: ‘;

and solve for x:
‘;
for (var i=0; i‘;
}
document.getElementById(‘equation_9114’).innerHTML = solution;
}

“;
solution = solution + ‘

Set y=0 and solve for x:
‘;
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_9114’).innerHTML = solution;
}

}

//init_mouse();

var c_9114=document.getElementById(“myCanvas_9114”);
var ctx_9114=c_9114.getContext(“2d”);

var change = 0.0001;

function create_lines_9114 () {
//draw line
//document.write(“hello world! “);
var polys = [];
polys.push(addPoly(1,6, 5));

// polys.push(addPoly(0.25, 1, 0));
// polys[1].color = ‘#8C8’;

return polys;
}

var polys_9114 = create_lines_9114();

var x1=xp(-10);
var y1=yp(1);
var x2=xp(10);
var y2=yp(1);
var dc_9114=0.05;

var t_9114 = 0;
var dt_9114 = 100;
//end_ct = 0;
var st_pt_x_9114 = 2;
var st_pt_y_9114 = 1;

var move_dir_9114 = 1.0; // 1 for up

var a_coeff_9114 = document.getElementById(“a_coeff_9114″);
a_coeff_9114.value = polys_9114[0].a+””;
var b_coeff_9114 = document.getElementById(“b_coeff_9114″);
b_coeff_9114.value = polys_9114[0].b+””;
var c_coeff_9114 = document.getElementById(“c_coeff_9114″);
c_coeff_9114.value = polys_9114[0].c+””;

//document.write(“test= “+c_coeff_9114.value+” “+polys_9114[0].c);
//setInterval(“draw_9114(ctx_9114, polys_9114)”, dt_9114);
draw_9114(ctx_9114, polys_9114);

a_coeff_9114.onchange = function() {
polys_9114[0].set_a(parseFloat(this.value));
draw_9114(ctx_9114, polys_9114);
}
b_coeff_9114.onchange = function() {
polys_9114[0].set_b(parseFloat(this.value));
draw_9114(ctx_9114, polys_9114);
}
c_coeff_9114.onchange = function() {
polys_9114[0].set_c(parseFloat(this.value));
draw_9114(ctx_9114, polys_9114);
}

[/script]

[/inline]

Hopefully, this can help students learn about factoring and quadratics in a more graphical way.

Notes:

This is my first interactive post. I use Javascript in combination with HTML5. Now I need to figure out how to interact with the image itself, instead of the textboxes.