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To find vertical asymptotes of a rational function, you need to locate where the function is undefined. This occurs where the denominator is zero. For the function \( f(x) = \frac{6x^2 - 15x + 6}{4x^2 - 10x} \), we solve \( 4x^2 - 10x = 0 \). Using factoring, split the equation into \( 2x(2x - 5) = 0 \). This factored form gives the solutions \( x = 0 \) and \( x = \frac{5}{2} \). These values, \( x = 0 \) and \( x = \frac{5}{2} \), are the vertical asymptotes where the function will tend to infinity. A vertical asymptote implies the curve approaches these lines but never crosses them. When graphing, expect to see steep inclines or declines as the curve approaches these vertical markings.

Finding horizontal asymptotes helps in understanding the end behavior of a rational function. They depict how the function behaves as \( x \) approaches extreme values in the positive or negative direction. In the case of our function, \( f(x) = \frac{6x^2 - 15x + 6}{4x^2 - 10x} \), both the numerator and the denominator have the same degree, which is 2.For rational functions where the degree of the numerator and denominator are equal, the horizontal asymptote can be found by dividing their leading coefficients. Here, this results in the horizontal asymptote \( y = \frac{3}{2} \). On a graph, the curve will flatten out and approach the line \( y = \frac{3}{2} \) as \( x \) becomes very large or very small.

The x-intercepts of a function are crucial points where the graph crosses the x-axis, meaning those where the output value, \( y \), equals zero. To find the x-intercepts of our function \( f(x) = \frac{6x^2 - 15x + 6}{4x^2 - 10x} \), we set the numerator \( 6x^2 - 15x + 6 = 0 \) and solve the quadratic equation.Applying the quadratic formula, or factoring if possible, will show that the solutions are \( x = 1 \) and \( x = \frac{1}{3} \). These intercepts indicate where the graph will touch or cross the x-axis. Remember that at x-intercepts, the y-value will exactly be zero.

Graphing rational functions can be greatly simplified by using graphing software. This technology allows you to set specific windows and view the graph clearly, revealing critical features and patterns. When setting up your graphing software for a function like \( f(x) = \frac{6x^2 - 15x + 6}{4x^2 - 10x} \), carefully choose a viewing window that captures all key behaviors without cutting off important parts of the graph.For example, based on the vertical asymptotes, x-intercepts, and the horizontal asymptote already discussed, a range for x and y from \([-1, 3]\) will neatly display these features. The advantage of graphing software is how instantly it can visualize complex functions, helping identify asymptotic behavior and intercepts at a glance. Adjust the window settings as needed to focus on regions of interest or to get a wider overview of the function's behavior.