Sketching the Graphs of Rational Equations 17 March 2011

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

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

HA: y = 0 VA: x = –1.5, 1 Holes: none

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

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: DNE (does not exist)

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 x-value y-value HA: y = 0 VA: x = –1.5, 1 Holes: none

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

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 = 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

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”.