Graphing Rational Functions
You will graph the function f(x) on the axes provided by completing the following steps.
1. Factor the function and find the domain restrictions. Simplify the function, if possible, and then find the vertical asymptote. Add these to the graph.
(Move the slider to change the number of asymptototes you need and then drag the line(s) to the correct spot)
2. Based on the degree of the numerator and denominator, what is the horizontal asymptote of the function? Add this to the graph.
(Move the slider to change the number of asymptototes you need and then drag the line(s) to the correct spot)
3. Does the function cross the horizontal asymptote? If so, where? Drag a point on the graph to this location.
4. Find the x and y intercepts. Drag them to the graph.
5. Look at the three vertical sections formed by the vertical asymptotes. Notice that we have already found important coordinates in each of the sections.
In the far left section, we know that the curve of our function must be completely contained in the bottom section of the coordinate plane where the x-intercept is, because it does not cross the horizontal asymptote there and it cannot cross the vertical asymptote. Sketch an approximate curve in the bottom left section that would approach both asymptotes.
6. In the middle of the graph, we know that the function will approach both vertical asymptotes. We already have a point because of the y-intercept, and we know that the function will not cross the horizontal asymptote here. The curve in the middle, then, must be a parabolic shape that opens up. Sketch that curve.
7. In the far right section, we have an intercept and the point where the function crosses the horizontal asymptote. The left side of this curve has to approach the vertical asymptote with values that are negative, go through the intercept, and then go through (8,1). Will it cross the horizontal asymptote again? No, we only found one point of crossing. So the curve must stay very close to the horizontal asymptote as x approaches infinity. Sketch this curve.