Sketch the graph of f(x)=(x^2 + 2)/x^2 - 4.
Fund the vertical and horizontal asymptotes, the critical points, the points of inflection and the concavity.
To sketch the graph of the function f(x) = (x^2 + 2)/(x^2 - 4), we will follow several steps to find the vertical and horizontal asymptotes, critical points, points of inflection, and determine the concavity. Let's break it down step by step:
Step 1: Find the Vertical Asymptotes
Vertical asymptotes occur when the denominator of the function is equal to zero. In this case, the denominator is (x^2 - 4). To find the vertical asymptotes, we need to solve the equation x^2 - 4 = 0 for x.
x^2 - 4 = 0
(x - 2)(x + 2) = 0
This equation has two solutions: x = 2 and x = -2. Hence, there are two vertical asymptotes at x = 2 and x = -2.
Step 2: Find the Horizontal Asymptote
To determine the horizontal asymptote, we check the degrees of the numerator and denominator polynomials.
In this case, the degree of the numerator polynomial (x^2 + 2) is 2, and the degree of the denominator polynomial (x^2 - 4) is also 2. Since the degrees of the numerator and denominator are equal, we need to examine the ratio of the coefficient of the highest degree term.
The highest degree term in both polynomials is x^2. The ratio of the coefficients is 1/1, which means the horizontal asymptote is y = 1.
Step 3: Find the Critical Points
To find critical points, we need to find the values of x where the derivative of the function f(x) is equal to zero or undefined.
First, let's find the derivative of f(x):
f'(x) = [(x^2 - 4)(2x) - (x^2 + 2)(2x)] / (x^2 - 4)^2
= (2x^3 - 8x - 2x^3 - 4x) / (x^2 - 4)^2
= (-12x) / (x^2 - 4)^2
To find the critical points, we need to solve the equation f'(x) = 0:
(-12x) / (x^2 - 4)^2 = 0
The numerator (-12x) is equal to zero when x = 0. Therefore, the critical point is located at x = 0.
Step 4: Find the Points of Inflection
To find the points of inflection, we need to find the x-values where the second derivative of the function f(x) changes sign or is equal to zero.
Taking the second derivative of f(x):
f''(x) = -12 / (x^2 - 4)^2 + (48x^2) / (x^2 - 4)^3
Setting the second derivative equal to zero and solving for x:
-12 / (x^2 - 4)^2 + (48x^2) / (x^2 - 4)^3 = 0
This equation doesn't have any real solutions, so there are no points of inflection.
Step 5: Determine Concavity
We can use the second derivative to determine the concavity of the graph. When f''(x) > 0, the graph is concave up, and when f''(x) < 0, the graph is concave down.
From our previous calculation,
f''(x) = -12 / (x^2 - 4)^2 + (48x^2) / (x^2 - 4)^3.
Since this equation is positive for all x, the graph of f(x) is concave up for all x.
Now that we have all the necessary information, it's time to sketch the graph of f(x) = (x^2 + 2)/(x^2 - 4), using the vertical and horizontal asymptotes, critical points, concavity, and the behavior of the function.