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 point(s) of the curve with equation y = (x^2+1)/(x^2-4), we need to find the points where the curve changes direction. The turning point(s) occur where the derivative of the equation is equal to zero.
(i) Finding the coordinates of the turning point(s):
1. Calculate the derivative of the equation:
y' = (2x(x^2 - 4) - (x^2 + 1)(2x))/(x^2 - 4)^2
= (2x^3 - 8x - 2x^3 - 2x)/(x^2 - 4)^2
= (-10x)/(x^2 - 4)^2
2. Set the derivative equal to zero and solve for x:
-10x = 0
x = 0
3. Substitute the x-coordinate into the original equation to find the corresponding y-coordinate:
y = (0^2 + 1)/(0^2 - 4)
= 1/(-4)
= -1/4
Therefore, the coordinate of the turning point is (0, -1/4).
(ii) Finding the equations of the asymptotes:
1. For the vertical asymptote(s), we need to determine the values of x for which the denominator of the equation is equal to zero (since division by zero is not defined). In this case, the denominator (x^2 - 4) equals zero when x = 2 or x = -2.
2. Therefore, we have two vertical asymptotes with equations x = 2 and x = -2.
3. For the horizontal asymptote, we need to determine the behavior of the curve as x approaches positive infinity and negative infinity.
As x approaches positive infinity, both the numerator (x^2 + 1) and denominator (x^2 - 4) of the equation grow without bounds. Thus, the horizontal asymptote is y = 1.
As x approaches negative infinity, both the numerator (x^2 + 1) and denominator (x^2 - 4) of the equation also grow without bounds. Hence, the horizontal asymptote is still y = 1.
(iii) Sketching the curve:
To sketch the curve, plot the coordinates of the turning point and the vertical asymptotes on a graph. Use the horizontal asymptote as a guide for the behavior of the curve towards infinity.
Keep in mind that the curve may have additional features, such as intercepts and specific regions of concavity, which can be determined by further analysis of the equation or by plotting additional points.