Here are the steps for graphing a rational function:
- Identify and draw the vertical asymptote using a dotted line.
- Identify and draw the horizontal asymptote using a dotted line.
- Plot the holes (if any)
- Find x-intercept (by using y = 0) and y-intercept (by x = 0) and plot them.
- Draw a table of two columns x and y and place the x-intercepts and vertical asymptotes in the table. Then take some random numbers in the x-column on either side of each of the x-intercepts and vertical asymptotes.
- Compute the corresponding y-values by substituting each of them in the function.
- Plot all points from the table and join them curves without touching the asymptotes.
Example: Graph the rational function f(x) = (x2 + 5x + 6) / (x2 + x – 2).
Solution:
We have already identified that its VA is x = 1, its HA is y = 1, and the hole is at (-2, -1/3). We use dotted lines for asymptotes so that we can take care that the graph doesn’t touch those lines. Note that, the simplified form of the given function is, f(x) = (x + 3) / (x – 1). Now, we will find the intercepts.
- For x-intercept, put y = 0. Then we get 0 = (x + 3) / (x – 1) ⇒ x + 3 = 0 ⇒ x = -3. So the x-intercept is at (-3, 0).
- For y-intercept, put x = 0. Then we get y = (0 + 3) / (0 – 1) ⇒ y = -3. So the y-intercept is at (0, -3).
We have the VA at x = 1 and x-intercept is at x = -3. Let us construct a table now with these two values in the column of x and some random numbers on either side of each of these numbers -3 and 1.
| x | y |
|---|---|
| -5 | y = (-5 + 3) / (-5 – 1) = 0.33 |
| -4 | y = (-4 + 3) / (-4 – 1) = 0.2 |
| -3 | 0 (x-int) |
| -2 | y = (-2 + 3) / (-2 – 1) = -0.33 |
| 0 | -3 (y-int) |
| 1 | VA |
| 2 | y = (2 + 3) / (2 – 1) = 5 |
| 3 | y = (3 + 3) / (3 – 1) = 3 |
Let us plot all these points on the graph along with all asymptotes, hole, and intercepts.
