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To find the x-intercepts of a rational function like \( f(x) = \frac{x^{2} - 3}{2x - 4} \), we need to determine where the function equals zero. This happens when the numerator is zero because a fraction is zero only if its numerator is zero (as long as the denominator isn't zero at the same time). For \( f(x) \), we set the numerator \( x^{2} - 3 \) equal to zero and solve for \( x \). This gives us the equation \( x^{2} - 3 = 0 \). Solving this, we find \( x = \pm \sqrt{3} \). These solutions indicate our x-intercepts are at \( x = \sqrt{3} \) and \( x = -\sqrt{3} \). Adding this to our graph, the function crosses the x-axis at two points: \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \).

To find the y-intercept of a rational function, we simply evaluate the function at \( x = 0 \). For the function \( f(x) = \frac{x^{2} - 3}{2x - 4} \), this means substituting 0 for \( x \):\( f(0) = \frac{0^{2} - 3}{2(0) - 4} = \frac{-3}{-4} = \frac{3}{4} \). The y-intercept is therefore at \( y = \frac{3}{4} \). On a graph, this is represented as the point \( (0, \frac{3}{4}) \). It’s important to always remember that the y-intercept represents where the graph crosses the y-axis, and this happens at \( x = 0 \).

Asymptotes in rational functions can be vertical or horizontal, and they represent lines that the graph approaches but never touches. For \( f(x) = \frac{x^{2} - 3}{2x - 4} \), vertical asymptotes occur where the denominator is zero, because the function becomes undefined there. Setting the denominator equal to zero: \( 2x - 4 = 0 \), we solve for \( x \) and get \( x = 2 \). Therefore, there is a vertical asymptote at \( x = 2 \). This forms a vertical line in the graph that the function approaches but never crosses.
Horizontal asymptotes depend on the degrees of the numerator and the denominator. Here, the numerator \( x^{2} \) has a degree of 2, and the denominator \( 2x \) has a degree of 1. Since the numerator’s degree is greater than the denominator’s, the function does not have a horizontal asymptote. Instead, the graph will increase or decrease without bound as \( x \) approaches infinity or negative infinity.

A rational function is any function represented as the quotient of two polynomials. In this context, \( f(x) = \frac{x^{2} - 3}{2x - 4} \) is a rational function because it is the division of the polynomial \( x^{2} - 3 \) by another polynomial \( 2x - 4 \). These functions can have interesting features such as intercepts and asymptotes which provide a lot of insight into their behavior.
The most crucial aspects of graphing rational functions involve finding these key features, which include:

• x-intercepts: where the graph crosses the x-axis
• y-intercept: where the graph crosses the y-axis
• vertical asymptotes: lines of x values where the function becomes undefined
• horizontal asymptotes: lines the function approaches as \( x \) tends to infinity or negative infinity

Understanding rational functions and their graphs help us visualize complex algebraic behaviors in an intuitive and graphical way. This makes it easier to predict the function’s values across its domain.