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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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| 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.
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| Photo by Math Principles in Everyday Life |