1. Sketching the Graphs of Rational Equations
17 March 2011

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…
…and not where the graph is.
We need points!
y-intercept
x-intercept(s)
Additional points

5. Solving for the y-intercept
Step 1: Rewrite the equation
Step 2: Substitute zero for x
Step 3: Solve for y
Step 4: Check that the y-intercept doesn’t happen at a discontinuity

6. HA: y = 0
VA: x = –1.5, 1
Holes: none

7. Solving for the x-intercept
Step 1: Rewrite the equation.
Step 2: Set the numerator equal to zero
Step 3: Solve for x
Step 4: Check that the x-intercept doesn’t happen at a discontinuity

8. HA: y = 0
VA: x = –1.5, 1
Holes: none

9. What if an intercept is impossible or matches a discontinuity?
Discard the solution!!!
y-int: DNE (does not exist)

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

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

12. Solving for Additional Points
Step 2: Use the table function in your calculator to generate the points
x-value
y-value -3 -2
.3333… -1
0.5
-1.75 2
.714 3
HA: y = 0
VA: x = –1.5, 1
Holes: none

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

14. 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

15. HA: y = 0
VA: x = 2
Holes: x = –1
y-int. = DNE
x-int. = – 0.5
x-value
y-value -6
-1.25 -5
-2.25 -3
1.75 -2
0.75

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

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