## Wednesday, March 27, 2013

### Graphical Sketch - Rational Function

Category: Analytic Geometry, Algebra

"Published in Suisun City, California, USA"

Sketch the graph for

Solution:

Consider the given equation above

Factor the numerator and denominator, if possible, we have

To get the x-intercept, we have to set the numerator to zero, as follows

The coordinates of the x-intercept is (2, 0).

To get the y-intercept, substitute x = 0 to the given equation as follows

The coordinates of the y-intercept is (0, 2).

To get the vertical asymptotes, we have to set the factors of the denominator to zero, as follows

If

If

The vertical asymptotes are x = 1 and x = -1.

To get the horizontal asymptote, consider the given equation above, as follows

Divide both the numerator and the denominator by the variable with the highest degree, we have

Take the limit of the given equation above as x approaches to infinity, we have

The horizontal asymptote is y = 0.

Next, we need to draw the dotted lines for vertical asymptotes and horizontal asymptote, we have

 Photo by Math Principles in Everyday Life

As you noticed that the horizontal asymptote contains the x-intercept. Usually, horizontal asymptotes never passes the curve or point but there are times that the horizontal asymptote will pass the curve.

The vertical asymptotes never passes the curve or point.

To start in sketching the curve, let's consider the given equation again

If x < -1, then

Draw the curve at the lower left side between the vertical and horizontal asymptotes.

If -1 < x < 0, then

From y-intercept, draw the curve upward approaching to vertical asymptote.

If 0 < x < 1, then

From y-intercept, draw the curve upward approaching to vertical asymptote.

If 1 < x < 2, then

From x-intercept, draw the curve downward approaching to vertical asymptote.

If x > 2, then

From x-intercept, draw the curve going to the right approaching to horizontal asymptote.

The final sketch of the graph for the given equation should be like this.

 Photo by Math Principles in Everyday Life

## Tuesday, March 26, 2013

### Centroid - Area, 2

Category: Integral Calculus, Analytic Geometry, Algebra, Physics, Mechanics

"Published in Newark, California, USA"

Find the centroid of the area bounded by two curves for

Solution:

The first thing that we have to do is to draw or sketch the two given curves using the principles of Analytic Geometry as follows

 Photo by Math Principles in Everyday Life

Next, we need to get their points of intersection by solving the two equations, two unknowns as follows

but

The above equation becomes

Equate each factor to zero and solve for the value of y. Therefore, y = 2 and y = -1.

Substitute the values of y either of the two equation in order to solve for the value of x, we have

If y = 2, then

If y = -1, then

Their points of intersection are (1, -1) and (4, 2).

Label further the figure and draw the horizontal strip, we have

 Photo by Math Principles in Everyday Life

The area bounded by the two curves is

The x value of the centroid for the figure bounded by two curves is given by the formula

If the length of a strip is x, then xC is ½ x. The above equation becomes

Therefore,

The y value of the centroid for the figure bounded by two curves is given by the formula

If the length of a strip is x, then yC is also equal to y which is the distance of a strip from x axis. Since dy is a very small measurement, then dy is negligible. The above equation becomes

Therefore,

Therefore, the coordinates of the centroid are