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Understanding the domain of a rational function is crucial for graphing it correctly. The domain includes all the possible values of 'x' that the function can accept. In the case of rational functions, like our example function \(h(x)=\frac{x^{2}-5x+4}{x^{2}-4}\), the domain is restricted by the denominator. We cannot have a denominator equal to zero because division by zero is undefined.

When we solve \(x^{2}-4=0\), we find that the values \(x=2\) and \(x=-2\) are excluded from the domain because they would make the denominator zero. Thus, the domain in interval notation is expressed as \((- \infty, -2) \cup (-2, 2) \cup (2, \infty)\). This means that the function can take any real number except 2 and -2.

To graph a function, identifying the intercepts is a fundamental step as they indicate where the graph crosses the axes. Let's start with x-intercepts which we find by setting the numerator of our function \(h(x)\) to zero. From the solution, we already know the x-intercepts are at the points (1,0) and (4,0).

The y-intercept is found by evaluating the function at \(x=0\). Doing this, we find the y-intercept at (0,-1). Notably, if there were multiple y-intercepts, this would be an indicator of a mistake as a function can only cross the y-axis once.

Vertical asymptotes are vertical lines that the graph of the function approaches but never touches or intersects. These occur where the function is undefined, typically at points where the denominator equals zero. For \(h(x)\), we've determined that there are vertical asymptotes at the points where \(x=2\) and \(x=-2\).

Visually, as the graph nears these lines, the function values either rise or fall indefinitely. Plotting vertical asymptotes on the graph provides a clear guide to the behavior of the graph and how it tends toward infinity at these critical values.

Horizontal asymptotes reflect the behavior of a function as the values of 'x' head towards infinity or negative infinity. They are horizontal lines that the function approaches. When the degrees of the numerator and the denominator of a rational function are the same, the horizontal asymptote equates to the ratio of their leading coefficients.

Since the degrees of both the numerator and denominator of our function are 2 and the leading coefficients are both 1, the horizontal asymptote is the line \(y=1\). This line indicates where the graph of the function settles as 'x' becomes very large in either the positive or negative direction.