What are the vertical and horizontal asymptotes for the function f(x)= 3x2/x2-4
1 answer:
Answer: f(x) will have vertical asymptotes at x=-2 and x=2 and horizontal asymptote at y=3.
Step-by-step explanation:
Given function:
The vertical asymptote occurs for those values of x which make function indeterminate or denominator 0.
i.e.
Hence, f(x) will have vertical asymptotes at x=-2 and x=2.
To find the horizontal asymptote , we can see that the degree of numerator and denominator is same i.e. 2.
So, the graph will horizontal asymptote at
i.e.
Hence, f(x) will have horizontal asymptote at y=3.
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Refer to Figure 1 for the diagram of the point and reference triangle needed to answer the problem.
Plot the point (2,-3) which is in quadrant 4
Draw a right triangle such that one leg is resting on the x axis and the other leg is perpendicular to the x axis. The hypotenuse goes from (0,0) to (2,-3).
The horizontal leg is 2 units long. So a = 2
The vertical leg is 3 units long. So b = 3
The hypotenuse is unknown but we can use the pythagorean theorem to find it. We'll call it c for now
Use Pythagorean theorem to find c
a^2 + b^2 = c^2
2^2 + 3^2 = c^2
4 + 9 = c^2
13 = c^2
c^2 = 13
c = sqrt(13)
The hypotenuse is sqrt(13) which is shown in the attached image (Figure 1)
Using figure 1, now refer to figure 2 to compute the ratios for sine, cosine and tangent
Figure 3 shows how to compute the ratios for cosecant, sectant, and cotangent
Note: With figure 1, the leg furthest from angle theta is the opposite side (in this case -3). The leg closest to the angle theta is 2 units long.
Plot the point (2,-3) which is in quadrant 4
Draw a right triangle such that one leg is resting on the x axis and the other leg is perpendicular to the x axis. The hypotenuse goes from (0,0) to (2,-3).
The horizontal leg is 2 units long. So a = 2
The vertical leg is 3 units long. So b = 3
The hypotenuse is unknown but we can use the pythagorean theorem to find it. We'll call it c for now
Use Pythagorean theorem to find c
a^2 + b^2 = c^2
2^2 + 3^2 = c^2
4 + 9 = c^2
13 = c^2
c^2 = 13
c = sqrt(13)
The hypotenuse is sqrt(13) which is shown in the attached image (Figure 1)
Using figure 1, now refer to figure 2 to compute the ratios for sine, cosine and tangent
Figure 3 shows how to compute the ratios for cosecant, sectant, and cotangent
Note: With figure 1, the leg furthest from angle theta is the opposite side (in this case -3). The leg closest to the angle theta is 2 units long.