91Ó°ÊÓ

Rational functions can have horizontal asymptotes, which show the behavior of the function as x approaches infinity or negative infinity.
For this problem, the horizontal asymptote is given as y = 3. This means that as x becomes very large, either positively or negatively, the values of the function f(x) will get closer and closer to 3.
This is useful because it helps us understand the long-term behavior of the function. To determine a horizontal asymptote, you often compare the degrees of the polynomials in the numerator and denominator.
For example, in the function provided, we have f(x) = \( \frac{3x^2}{x^2 - 1} \). The degrees of the numerator and denominator are both 2, making the ratios of their leading coefficients (3/1) the horizontal asymptote y = 3.

Vertical asymptotes are vertical lines where the function approaches infinity and are found where the denominator of the rational function equals zero.
In this problem, the vertical asymptotes occur at x = -1 and x = 1. This means that as x gets close to either -1 or 1, the value of the function will grow very large in positive or negative direction.
To find vertical asymptotes, set the denominator equal to zero and solve for x. For the function \( f(x) = \frac{3x^2}{x^2 - 1} \), the denominator is x^2 - 1.
Setting it to zero, you get x^2 - 1 = 0, which solves to x = ±1. Hence, those are your vertical asymptotes. These asymptotes tell you where the function is undefined and help in sketching the graph accurately.

A function can have symmetry, making it easier to sketch and understand its behavior.
For this particular rational function, it is symmetric with respect to the y-axis. This means that the function is even, and its formula uses only even powers of x.
The given function is \( f(x) = \frac{3x^2}{x^2 - 1} \). Notice that all the variables x are raised to an even power (x^2).
Mathematically, a function is even if f(x) = f(-x). This symmetry helps simplify graphing because you only need to plot one half of the graph and mirror it across the y-axis.
Recognizing symmetry can save time and ensure the graph looks accurate.

Intercepts are the points where the graph of a function crosses the x-axis or y-axis.
For the given function, both x-intercept and y-intercept occur at the origin (0,0).
To find x-intercepts, set the numerator of the rational function to zero and solve for x. In our function \( f(x) = \frac{3x^2}{x^2 - 1} \), setting the numerator 3x^2 to zero results in x = 0. Hence, (0,0) is an x-intercept.
To find y-intercepts, set x = 0 and solve for y. Using the function formula \( f(0) = \frac{3(0)^2}{(0)^2 - 1} = 0 \), you get y = 0. Thus, (0,0) is also the y-intercept.
Intercepts give precise points to plot on the graph, providing a clear start to sketching the function.