Based on my students’ feedback, I’ve made it so that when you change the equation of the line the movement animates. This makes it much easier to see what happens when, for example, you change the slope of a line.
P.S. You can also turn off the interactivity if you just want to show a simple graph. y = x2-1 is shown below:
If you recall, straight lines have a general equation that looks like this:
This is called the slope-intercept form of the equation, because m gives the slope, and b tells where the line intercepts the y-axis. For example the line:
Now, in the slope-intercept form, m and b represent numbers. In our example, m = 2 and b = 3.
So what if, instead of calling them m and b we used the same letter (let’s use a) and just gave two different subscripts so:
therefore equation (1):
Now, in case you’re wondering why we picked m = a1 instead of m = a0, it’s because of the exponents of x. You see, in the equation x has an exponent of 1, and the constant b could be thought to be multiplying x with an exponent of 0. Considering this, we could rewrite our equation of the line (2):
So in equation (3) the subscript refers to the exponent of x.
Now we can expand this a bit more. What if we had a term with x2 in an equation:
Now we have three coefficients:
And the graph would look like this.
Because of the x2 term (specifically because it has the highest exponent in the equation), this is called a second-order polynomial — that’s why the graph above has a little input box where the order is 2. In fact, on the graph above, you can change the order to see how the equation changes. Indeed, you can also change the coefficients to see how the graph changes.
A second order polynomial is a parabola, while, as you’ve probably guessed, a first order polynomial is a straight line. What’s a zero’th order polynomial?
Finally, we can write a general equation for a polynomial — just like we have the slope-intercept form of a line — using the a coefficients like:
You can use the graphs to tinker around and see what different order polynomials look like, and how changing the coefficients change the graphs. I sort-of like the one below:
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).
Starting point: (x,y) = ( , )
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:
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.
where the point (h, k) is the location of the vertex of the parabola. In the example above, h = 1 and k = 2.
To translate between the two forms of the equation, you have to rewrite them. Start by expanding the vertex form:
y = a(x - h)2 + k
y = a(x - h)(x - h) + k
multiplied out to get:
y = a(x2 - 2hx + h2) + k
now distribute the a:
y = ax2 - 2ahx + ah2 + k
finally, group all the coefficients:
y = (a)x2 - (2ah)x + (ah2 + k)
This equation has the same form as y = ax2 + bx + c if:
Vertex to Standard Form:
a = a
b = -2ah
c = ah2+k
And we can rearrange these equations to go the other way, to find the vertex form from the standard form:
Standard to Vertex Form:
In sum, you can write the standard equation for a parabola as:
And you can write the equation for the same parabola in vertex form as:
It's a more interactive version of the Excel parabola program in that you can move the curve by dragging on some control points, rather than just having to enter the coefficients of the equation. The program is still in development, but it is at a useful stage right now, so I thought I'd make it available for anyone who wanted to try it.
The program is fairly straightforward to use. You can move the curve (translate it) up and down, and expand or tighten the area within the parabola.
The program also displays the equation of the curve in standard form:
What the buttons do.
I'm also working making the standard equation editable by clicking on it and typing, and am considering showing the x-axis intercepts, which will give algebra students a nice, visual way to of checking their factoring.