Sketching the Graphs of Rational Equations

Consider the equation below: Solve for the discontinuities.

4x

2xy

Your Turn:

Solve for the discontinuities of problems 1 – 6 on Sketching the Graphs of Rational Equations – Part I

Answers: HA: y = 1 VA: x = 2 Holes: DNE

HA: y = –½ VA: x = –3 Holes: DNE

HA: y = 2 VA: x = 1 Holes: x = –2

HA: y = 0 VA: x = 2 Holes: DNE

HA: y = 0 VA: x = 2 Holes: x = –1

HA: y = 2 VA: x = –3 Holes: x = 0

Summary – 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

Leave Blank for Now…

Example #14x

2xy

Example #22x

10x7xy

2

Your Turn:

For problems 1 – 6, solve for the y-intercept. Check your answers in your graphing calculator!!!

Answers:1. y-int = –3

2. y-int = –⅔

3. y-int = –6

4. y-int = –1.5

5. y-int = –½

6. y-int = DNE

HA: y = 2VA: x = –3Holes: x = 0

x3x

x2y

2

2

#6

Solving for the y-intercept Step 1: Rewrite the equation Step 2: Substitute zero for x Step 3: Solve for y

If the y-intercept is undefined or indeterminate, then the y-int. is DNE!!!

HA: y = 0VA: x = 0Holes: DNE

2x

2xy

AdditionalExample #1:

HA: y = 1VA: x = 5Holes: x = 0

x5x

x2xy

2

2

AdditionalExample #2:

Solving for the x-intercept(s)

Step 1: Rewrite the equation Step 2: Substitute zero for y Step 3: Solve for x

Leave Blank for Now… Step 4: Leave Blank for Now…

Example #14x

2xy

Your Turn:

For problems 1 – 4, solve for the x-intercept(s). Check your answers in your graphing calculator!!!

Answers:1. x-int = –6

2. x-int = –4

3. x-int = –3

4. x-int = DNE

HA: y = 2VA: x = 1Holes: x = –2

2xx

12x10x2y

2

2

#3

HA: y = 0VA: x = 2Holes: DNE

2x

3y

#4

Solving for the x-intercept(s) Step 1: Rewrite the equation Step 2: Substitute zero for y Step 3: Solve for x

If the answer is impossible, then the x-intercept is DNE

Step 4: Check if the x-intercept matches any of the discontinuities. If it does, REJECT that x-intercept!!!!

Your Turn:

Solve for the x-intercept(s) of problems 5 – 6.

HA: y = 0VA: x = 2Holes: x = –1

2xx

1xy

2

#5

HA: y = 2VA: x = –3Holes: x = 0

x3x

x2y

2

2

#6

4 m – Solve for the y-int. and the x-int.

Finding Additional Points We can use our graphing calculators to find

additional points! Step 1: Make a table that has two points before

and after each VA and hole. Step 2: Type the equation into y1 of graphing

calculator. Step 3: Use the table function to find points to

fill into the table. Pick points that are easy to graph!!!

#1

HA: y = 1 VA: x = 2 Holes: DNE

2x

6xy

#3

HA: y = 2 VA: x = 1 Holes: x = –2

2xx

12x10x2y

2

2

Your Turn:

On the “Sketching the Graphs of Rational Equations – Part I” handout, make a table of additional points for problems 2, 4, 5, and 6.

Sketching – Putting It All Together!!! Step 1: Graph the HAs and VAs

Remember, we use dashed lines to represent asymptotes!

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 Step 5: Graph any holes

HA: y = 1

VA: x = 2

Holes: none

y-int. = –3

x-int. = –6

x-values y-values

0 –3

1 –7

2 Error

3 9

4 5

2x

6xy

#1

#3

HA: y = 2 VA: x = 1 Holes: x = –2

2xx

12x10x2y

2

2

x-values y-values

-7 1

-3 0

-2 error

-1 -2

0 -6

1 error

2 10

3 6

Your Turn:

On the “Sketching the Graphs of Rational Equations – Part I” handout, sketch the graphs of problems 2, 4, 5, and 6.

Homework

Create a table for and sketch the graphs of problems #4 - #6 on “Sketching the Graphs of Rational Equations – Part II”.

#4

HA: y = 2VA: x = 2Holes: DNE

y-int. = -5x-int. = -5

#5

HA: y = 0VA: x = 2Holes: x = 7

y-int. = -½ x-int. = DNE

#6

HA: y = 1VA: x = -2Holes: x = 3

y-int. = 3x-int. = -6