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In the study of rational functions, asymptotes play a crucial role in understanding the behavior of graphs. An asymptote is a line that the graph of a function approaches but never actually reaches. There are two main types of asymptotes relevant to rational functions:

• Vertical Asymptotes: These occur where the denominator of the function is zero, leading to division by zero, which is undefined. For the function \( f(x) = \frac{2x^2}{x^2 - 1} \), the vertical asymptotes are found by setting the denominator equal to zero: \( x^2 - 1 = 0 \). Solving this gives \( x = \pm 1 \), indicating vertical asymptotes at these points.
• Horizontal Asymptotes: These describe how the function behaves as \( x \) approaches infinity. For the function above, both the numerator and denominator have the same degree (2). Therefore, we find the horizontal asymptote by dividing the leading coefficients: \( \frac{2}{1} = 2 \). This tells us that as \( x \to \pm \infty \), the function approaches \( y = 2 \).

Critical points help identify the locations where the graph of a function changes direction or has a peak or trough. These are found by setting the first derivative equal to zero or finding where the derivative does not exist.

For our function \( f(x) = \frac{2x^2}{x^2 - 1} \), we take its derivative and set it to zero: \( f'(x) = \frac{-4x}{(x^2 - 1)^2} = 0 \). The only solution to this is \( x = 0 \), where the derivative equals zero, indicating a critical point. Additionally, critical points include locations where the derivative is undefined. Here, this occurs at \( x = \pm 1 \), highlighting these as places where the behavior of the graph changes significantly, aligning with the vertical asymptotes.

Derivatives are fundamental in calculus, as they measure how a function changes at any given point, essentially describing the slope of the function. To find the derivative of a rational function, the quotient rule is used.

For the function \( f(x) = \frac{2x^2}{x^2 - 1} \), the derivative \( f'(x) \) is calculated using the quotient rule formula \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \). Here, \( u = 2x^2 \) and \( v = x^2 - 1 \). Applying the quotient rule gives us:

• \( u' = 4x \)
• \( v' = 2x \)

Plugging these into the formula, we find:\[ f'(x) = \frac{(4x)(x^2 - 1) - (2x^2)(2x)}{(x^2 - 1)^2} = \frac{-4x}{(x^2 - 1)^2} \].
The derivative helps us understand the problem's critical points and informs the graph's increasing or decreasing nature.

A sign diagram is a valuable tool used to determine where a function is increasing or decreasing. This is especially useful once we know the derivative, as the sign of the derivative tells us about the behavior of the function.

For the function \( f(x) = \frac{2x^2}{x^2 - 1} \), the derivative is \( f'(x) = \frac{-4x}{(x^2 - 1)^2} \). To create a sign diagram, we test intervals between critical points and asymptotes to see if the function is increasing or decreasing:

• For \((-\infty, -1)\), testing values like \( x = -2 \) shows \( f'(-2) > 0 \), indicating the function is increasing here.
• For \((-1, 0)\), \( x = -0.5 \) results in \( f'(-0.5) < 0 \), meaning the function is decreasing in that interval.
• Similarly, between \((0, 1)\) with \( x = 0.5 \), the function is increasing as \( f'(0.5) > 0 \).
• Lastly, for \((1, \infty)\), testing \( x = 2 \) gives \( f'(2) < 0 \), showing the function decreases.

By piecing together these intervals, a comprehensive picture of the function's graph emerges, showing where it rises, falls, and reaches extremum points.