1. Sketching the Graphs of Rational Equations 18 November 2010

2. Consider the equation below: What are its discontinuities? HA: y = 0 VA: x = -1.5, 1 Holes: none

3. What We Know How To: Identify discontinuities Algebraically solve for discontinuities Tell the difference between vertical asymptotes and removable discontinuities

4. But aren’t we missing something? But discontinuities represent where the graph isn’t… …but not where the graph is. We need points!  y-intercept  x-intercept(s)  Additional points

5. Solving for the y-intercept Step 1: Substitute zero for x Step 2: Solve for y Step 3: Check that the y-intercept doesn’t happen at a discontinuity HA: y = 0 VA: x = -1.5, 1 Holes: none

6. Solving for the x-intercept Step 1: Set the numerator equal to zero Step 2: Solve for x Step 3: Check that the x- intercept doesn’t happen at a discontinuity HA: y = 0 VA: x = -1.5, 1 Holes: none

7. What if an intercept is impossible or matches a discontinuity? Discard the solution!!! y-int: none

8. Your Turn: On the “Sketching the Graphs of Rational Equations – Part I” handout, solve for the x- intercept(s) and the y-intercept.

9. Solving for Additional Points Step 1: Make a table that has two points before and after each VA and hole. HA: y = 0 VA: x = -1.5, 1 Holes: none x-valuey-value -3 -2 0.5 2 3

10. Solving for Additional Points, cont. Step 2: Substitute x- values from the table into the equation, and solve for y.

11. Solving for Additional Points, cont. Step 3: Complete the table x-valuey-value -30 -2.3333… 0.5-1.75 2.714 3.3333…

12. Your Turn: On the “Sketching the Graphs of Rational Equations – Part I” handout, make a table of additional points.

13. Sketching – Putting It All Together!!! Step 1: Graph all the discontinuities (HAs, VAs, and holes)  Remember, we use dashed lines to represent asymptotes and open circles to represent holes! Step 2: Graph the y-intercept and the x- intercept(s) (if they exist) Step 3: Graph the points from the table Step 4: Connect the points with lines

14. HA: y = -.25 VA: x = -4 Holes: none y-int. = 0.25 x-int. = 4 x-valuey-value -6-1.25 -5-2.25 -31.75 -20.75

15. Your Turn: On the “Sketching the Graphs of Rational Equations – Part I” handout, sketch the graphs of the equations.

16. Homework Finish “Sketching the Graphs of Rational Equations – Part II”.