Sketching the Graphs of Rational Equations 18 November 2010

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

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

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

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

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

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

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

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

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

Solving for Additional Points, cont. Step 3: Complete the table x-valuey-value … …

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

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

HA: y = -.25 VA: x = -4 Holes: none y-int. = 0.25 x-int. = 4 x-valuey-value

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

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