__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

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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 |