Thursday, June 30, 2011

Parametric

A curve can be written in normal equation involving x and y (in 2D), such as y = ln (x) + 2. However, the curve can be described in parametric equations as well. A pair or a set of parametric equations may look like the equations below:


System Normal Equation Parametric Equations
1.
2D
x = 2
2.
2D
3.
2D
4.
3D


Let's see the process of graphing the parametric equations:

  1. If we draw x = 2, we will just draw a vertical line that passes through the point (2,0) regardless with the direction. Somehow, in this parametric equations, we find that the line was built from bottom up as t increases. Click the 'play' button at the bottom left corner to see the forming of this line.



  2. For the second equation , normally we would just draw the ellipse from any starting point regardless of the orientation. In parametric equations , the starting point is fixed relying on the given range of t, same with its orientation. In the graph below, click the 'play' button at the bottom left corner of the graph and we can see the ellipse of this parametric equations starts from the point (x,y) = (2,0) and moving counterclockwise.



  3. For and its parametric equations , the graph of the paramteric equations is drawn as how we draw for the normal equation. Note that the set of parametric equations does not always use the variable t. Using other variables are fine as well.



  4. For a 3D surface, , its parametric equations can be written in one or two variables, depends. In this example, I give a set of parametric equations of x, y and z to be written in two variables s and t, .


Can you try to build one set of parametric equations from the set of equations {z = ln(y), x = 3} and figure out the graph of the set of equations?


Betty, Created with GeoGebra

Saturday, June 25, 2011

Parabola

A parabola has a standard equation of

------(1)

or
------(2)

where (h,k) is the coordinate of the vertex (labeled as V in the graph).

For the parabola of type (1), the parabola is either opening upwards (p > 0) or downwards (p < 0). This type of parabola has a focus at F(h, k + p) and a directrix at y = k - p.


On the other hand, for the parabola of type (2), the parabola is either opening rightwards (p > 0) or leftwards (p < 0). This type of parabola has a focus at F(h + p, k) and a directrix at x = h - p.

Betty, Created with GeoGebra

Hyperbola

A hyperbola which is centered at (h,k) has a standard equation

------(1)

if the hyperbola is opening sidewards, else, if it's opeing upwards and downwards, the standard equation will be
------(2)


Let's take a look at equation type (1):


This hyperbola in blue has
a) vertices at B1 and B2 at ,
b) foci at F1 and F2 at where ,
c) asymptotes in red dotted lines .


Now, let's look at equation type (2):


This hyperbola in blue has
a) vertices at B1 and B2 at ,
b) foci at F1 and F2 at where ,
c) asymptotes in red dotted lines .

Betty, Created with GeoGebra

Ellipse

An ellipse has a standard equation of



where
(h,k) is the center of the ellipse,
a is the distance between the vertex and the center, and
b is the distance between the endpoint of minor axis and the center.

Look at the graph below, point A is the center.
When a > b, points B1 and B2 are the vertices, points C1 and C2 are the endpoints of minor axis, and F1 and F2 are the foci.
When a < b, points C1 and C2 are the vertices, points B1 and B2 are the endpoints of minor axis, and F3 and F4 are the foci.
When a = b, a circle is formed. As such, we can say that circle is a special type of ellipse.

The line segment which connects the vertices is called as major axis. The line segment which is shorter, perpendicular to the major axis and passes through the center is called the minor axis.

You may slide the values of a, b, h or k to observe the changing of the graph.



Betty, Created with GeoGebra

Saturday, June 4, 2011

Indefinite Integrals

Let say we have a function (x). If we integrate (x) with respect to x, we get the antiderivative F(x) + C as follow:
where C is a constant.

Now, consider you do not know the function (x) but you need to sketch the graph of F(x) from the graph of (x) only, taking C = 0. How are you going to start?

Let's take a look at the graph below first:



The blue curve the curve for (x) and the maroon curve is the curve of F(x).

Line L is the tangent at point B, with equation y = mx + c, where m is the slope of tangent and c is the y-intercept of the tangent.

Point A is a point at curve f (x) and point B is the corresponding point at curve F(x) having the same x-coordinate of point A.

Now, try to move point A and observe the following:
(a) As point A falls above x-axis, curve F(x) is increasing.
(b) As point A falls on the x-axis, curve F(x) reaches to its extremum (maximum/minimum).
(c) As point A falls below the x-axis, curve F(x) is decreasing.
(d) As point A falls on the extremum point, point B is a point of inflection, which the curve F(x) changes its concavity.

Take note that the y-coordinate of point A is exactly the slope of tangnet (m) of point B!

Perhaps you can try to draw any new curve f(x) and find its integral F(x), taking C = 0?


Betty, Created with GeoGebra