91Ó°ÊÓ

Understanding the domain is crucial when dealing with rational functions. The domain includes all possible input values that result in a well-defined function, meaning they don’t produce a division by zero. To find the domain of a rational function like \(h(x)=\frac{x^{2}-5x+4}{x^{2}-4}\), we look at the denominator and determine when it equals zero, since division by zero is undefined.

In this case, if you factor the denominator as \((x-2)(x+2)\), you can see that \(x\) can be any number except 2 and -2. Thus, the domain of \(h(x)\) is all real numbers except for 2 and -2.

The x-intercepts and y-intercepts provide us with crucial information about where a graph crosses the x-axis and y-axis. To find the x-intercepts of \(h(x)\), we set the numerator equal to zero, since this is where the output of the function will be zero. In the example \(x^{2}-5x+4=0\), the x-intercepts are found to be at 1 and 4 after solving the equation.

For the y-intercept, we substitute \(x = 0\) in \(h(x)\), thus the y-intercept is simply \(h(0)\). In our case, it’s \(-1\), so the function crosses the y-axis at \((0, -1)\). These intercepts provide anchor points for the graph.

Asymptotes represent lines that the graph approaches but never touches. Vertical asymptotes occur where the function is undefined, which is typically where the denominator is zero. In our example, we already know that the function is undefined at \(x = 2\) and \(x = -2\), hence these are the vertical asymptotes.

Horizontal asymptotes are a bit different. To find them, we compare the degrees of the polynomials in the numerator and the denominator. When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Here, with both the numerator and denominator having a leading term of \(x^2\), the horizontal asymptote is at \(y = 1\), which is the ratio of the coefficients of these terms.

After identifying intercepts and asymptotes, a key step in graphing rational functions is to plot additional points to refine the graph. Choose some values of \(x\) on either side of the vertical asymptotes and calculate \(h(x)\) for these points. These points help us visualize how the graph behaves near the intercepts and asymptotes.

For instance, when \(x\) is just smaller than -2 and just larger than 2, you'd expect the graph to drop off towards negative infinity as it nears the vertical asymptotes, if the function is negative on that interval. Elsewhere, the function will follow a path that respects the established horizontal asymptote. Plotting points in other intervals helps to shape the graph between our known reference points (intercepts and asymptotes), providing a more accurate sketch.