for the curve with equation y=(x^2+1)/(x^2-4), find (i) the cordinates of the turning point(s) (ii) the equations of the asymptotes (iii) sketch the curve
To find the coordinates of the turning points, we need to find the critical points of the curve, which occur where the derivative is equal to zero or does not exist.
Step 1: Find the derivative of the equation y = (x^2+1)/(x^2-4).
Using the quotient rule, we get:
y' = [(x^2-4)(2x)-(x^2+1)(2x)] / (x^2-4)^2
Simplifying, we have:
y' = (2x^3 - 8x) / (x^2-4)^2
Step 2: Set the derivative equal to zero and solve for x.
2x^3 - 8x = 0
Factor out 2x:
2x(x^2-4) = 0
This gives us two solutions:
x = 0 and x = ±2.
Step 3: Find the corresponding y-coordinates for the turning points.
Substitute the values of x in the original equation:
For x = 0:
y = (0^2+1) / (0^2-4) = 1/(-4) = -1/4
So, one turning point is (0, -1/4).
For x = 2:
y = (2^2+1) / (2^2-4) = 5/0
Here, the denominator is zero, indicating that there is a vertical asymptote at x = 2. Therefore, there is no turning point at x = 2.
For x = -2:
y = ((-2)^2+1) / ((-2)^2-4) = 5/4
So, another turning point is (-2, 5/4).
(i) The coordinates of the turning points are (0, -1/4) and (-2, 5/4).
To find the equations of the asymptotes, we consider the behavior of the function as x approaches infinity and negative infinity.
(ii) Vertical Asymptote:
As x approaches infinity or negative infinity, the value of x^2 in the equation becomes dominant. Therefore, we can approximate the equation as y ≈ x^2 / x^2 = 1, which means there is a horizontal line at y = 1. So, the horizontal asymptote is y = 1.
(iii) Horizontal Asymptote:
To find the horizontal asymptote, we take the limit of the equation as x approaches infinity or negative infinity.
lim(x→±∞) [(x^2+1)/(x^2-4)]
Since the degree of the numerator and denominator is the same (both 2), we look at the ratio of the leading coefficients:
lim(x→±∞) [1/1] = 1
Therefore, there is no horizontal asymptote.
To sketch the curve, plot the turning points at (0, -1/4) and (-2, 5/4). Also, draw the vertical asymptote x = 2. Since there is no horizontal asymptote, the curve can extend infinitely in the y-direction.