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Graph sketching involves understanding the overall shape of a function on a coordinate plane. Rational functions can be a bit tricky because of their asymptotes and intercepts. Here are some steps to help you sketch these graphs:

• Find the **intercepts** by setting the numerator equal to zero for x-intercepts and sub in zero for y-intercepts.
• Identify **vertical asymptotes** by setting the denominator to zero.
• Determine **horizontal or slant asymptotes** based on the degree of the numerator and denominator.
• Plot these key features on the graph and understand the behavior of the function around these points.

Break down the function into simpler parts and consider how the curve behaves as it approaches the asymptotes. Smoothly connect calculated points to better visualize the function. In our exercises, the most critical points, like intercepts and asymptotes, act as the skeleton for our rational function graphs.

Intercepts are points where the graph of a function crosses the axes. In rational functions, they are essential in setting up your graph's initial structure.

X-Intercepts

For rational functions, x-intercepts are found by setting the numerator equal to zero. However, not all numerators have real solutions, as seen in our examples. This lack of x-intercepts affects how the graph is structured since it never touches the x-axis.

Y-Intercepts

These are easier to find. Simply set x to zero in the function. For example, for the function \( y = \frac{4x^2 + 1}{x^2-1} \), substituting x gives a y-intercept of -1. Likewise, for \( y = \frac{4x^2 + 1}{x^2+1} \), it yields a y-intercept of 1. Here, the graph will cross the y-axis at these points.

Asymptotes provide insights into the infinite behavior of rational functions. Understanding them helps in comprehending how the graph behaves over larger values of x.

Vertical Asymptotes

These occur where the denominator equals zero, creating a division by zero scenario that the graph cannot cross. It predicts where the function will go to infinity in either the positive or negative direction. For \( y = \frac{4x^2 + 1}{x^2-1} \), the vertical asymptotes occur at x = ±1.

Horizontal Asymptotes

The horizontal asymptote is the constant value that the graph tends to as x approaches infinity. You find it by comparing the degrees of the numerator and denominator. With equal degrees, take the ratio of the leading coefficients. In both examples \( y = \frac{4x^2 + 1}{x^2-1} \) and \( y = \frac{4x^2 + 1}{x^2+1} \), the horizontal asymptote is y = 4.These lines provide a boundary that the graph of the rational function will approach but never cross.