Sketching the Graphs of Rational Equations

18 November 2010

Consider the equation below:

What are its discontinuities?

)1)(32(

3

xx

xy

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

13

3

)1)(3(

3

)1)(30(

3

)10)(302(

30

)1)(32(

3

y

y

y

y

y

xx

xy

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

3

03

)1)(32(

3

x

x

xx

xy

HA: y = 0

VA: x = -1.5, 1

Holes: none

What if an intercept is impossible or matches a discontinuity?

!!!0

120

12000

1200

0:12

2

2

y

y

y

xVAx

xxy 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-value y-value

-3

-2

-1

0.5

2

3

Solving for Additional Points, cont. Step 2: Substitute x-

values from the table into the equation, and solve for y.

012

0

)4)(3(

0

)4)(36(

0

)13)(332(

33

)1)(32(

3

y

y

y

y

y

xx

xy

Solving for Additional Points, cont. Step 3: Complete the

tablex-value y-value

-3 0

-2 .3333…

-1 -1

0.5 -1.75

2 .714

3 .3333…

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

164

4

x

xy

HA: y = -.25

VA: x = -4

Holes: none

y-int. = 0.25

x-int. = 4

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”.