alg. 2 honors unit 6...

Alg. 2 Honors – Unit 6 Notes

Day 1 – Simplify and Multiply Rational Expressions

Simplified form of a rational expression -

Simplify the following rational expressions.

1. 30

42 2.

3x 12

3

3.

4 2

3

18x y z

9xy z 4.

2

2

4y 8y

y 4

5. 2

4x 20

x 11x 30

6.

4 15

5 16 7.

x 4 6

3(x 5) x(x 4)

8. 2

2 2

2x 10 9x

x 3x 10 6x 12x

9.

2 2

2

5k 10k 6k 6

2k 2 15k 30

10.

2 2

2

3y 12 4y 400

12y 120 y 8y 20

11. 2

3

x 4x x 1

x 1

12.

4 3 22

3

2x 12x 18xx 3x 9

x 27

Day 2 – Divide Rational Expressions


1. 3 9

8 16 2.

2

a 2 a

3a 6 a 4

3.

2

2 2

7x 21x

x 3x x 4x 21

4. 25y 20y 16

20y

5.

2 2

2 2

k m 6k 6m

k 2km 3m 3

6. 2

2

3 8x 8x

x 7 x 6x 7

7.

2

2 2

k 5 2k 11k 5

9k 18k 2k 5k 2

Day 3 – Add and Subtract Rational Expressions


1. 11 7

12 12 2.

3a 1 a 5

4 4

3.

2y 1

y 1 1 y

4. 2x 3x 10

x 5 5 x

5.

5 1

6 3 6.

5 3

3x 4

7. 2

4 6

4m m

8.

x 1

x 1 x

9.

m 4

m 2 m 2

10. 2 2

4 x 1

4x 2x 4x

11.

2 2

x 1 6

x 6x 9 x 9

Day 4 – Complex Fractions

Complex fractions -


1.

8

5xx 1 4

15 15

2.

2x

x 11 2

3x x 1

3.

6y y

4 48 3y

2 2

4. 2

2

5 7

2x x3 1

x 2x

Day 5 – Solve Rational Equations

Solve the following rational equations by cross multiplying.

1. 20 5

3x 5 x 2

2.

6 12

x 2 x 1

STEPS TO SOLVE EQUATIONS CONTAING MULTIPLE RATIONAL EXPRESSIONS

1) Get a common denominator for the entire equation (both sides).

2) Eliminate the bottom and solve the equation that was left from the numerators.

3) Check to see if any of your solutions make the denominators of your problem undefined.

Solve the following rational equations

3. 2a 3 2a 1

6 3 2

4. 5x 1

5x 1 x

5. 2y 3 12

2y 3 y 3

6. 2

3x 5 82

x 3 x x 12

7. 2

m 3 2m1

m 1 m 1

8. 2

2

6x 3x 4

x 16 x 4 x 4

Day 6 – Graph Simple Rational Functions

Rational function - Asymptote -

STEPS TO GRAPH RATIONAL FUNCTIONS

Find the vertical asymptotes, if any.

Find the horizontal asymptotes, if any.

Find the holes, if any.

Find the x-intercepts, if any.

Use a table and pick at least 2 x-values on each side of the vertical asymptotes to fill in your table and

plot these points on your graph.

Sketch the graph and use your graphing calculator to check.

How to find asymptotes, holes and x-intercepts…

Given the function p(x)

f (x)q(x)

Vertical Asymptote:

To find the V.A. set equal to and solve.

Horizontal Asymptote:

To find the H. A. there are three cases…

o Case 1: The degree of is the degree of .

The H.A. is .


The H.A. is , where and are leading

coefficients of the numerator and denominator


Then the graph has .

Holes:

Holes are created when are . Set the factors equal to zero

and solve.

x-intercepts:

To find the x-intercepts set equal to and solve.

x y

x y

x y

x y

1. 3

yx

2. 4

1

xy

x

Vertical Asymptotes: Vertical Asymptotes:

Horizontal Asymptotes: Horizontal Asymptotes:

Holes: Holes:

x-intercepts: x-intercepts:

3. 1

23

yx

4. 2

2

9

7 12

xy

x x



Holes: Holes:


x y

x y

Day 7 – Graph General Rational Functions

STEPS TO GRAPH RATIONAL FUNCTIONS

Find the vertical asymptotes, if any.

Find the horizontal asymptotes, if any.

Find the holes, if any.

Find the x-intercepts, if any.

Use a table and pick at least 2 x-values on each side of the vertical asymptotes to fill in your table and

plot these points on your graph.

Sketch the graph and use your graphing calculator to check.

Graph the following rational equations.

1. 2

4

25

y

x 2.

2 1

xy

x



Holes: Holes:


x y

x y

x y

x y

3. 2

2

3 6

( 16)( 2)

x xy

x x 4.

2

2

2

12

xy

x x



Holes: Holes:


5. 2

50 100

( 25)( 2)

xy

x x 6.

2

2

4

4 4

xy

x x



Holes: Holes:


x y

x y

x y

x y

Day 8 – Graph Rational Functions with Slant Asymptotes

Slant Asymptotes –

1. 2

5

1

y

x 2.

2

2

16

4

xy

x



Holes: Holes:


3. 2

1

xy

x 4.

3

2

9

4

x xy

x



Holes: Holes:


x y

Day 9 Graphing Rational Functions

The Parent Rational Function f(x) = 1

𝑥

Vertical Asymptote:

Horizontal Asymptote:

Sketch the graphs of these functions on your graphing calculator and describe the transformation.

f(x) = −1

𝑥 f(x) =

3

𝑥 f(x) =

1

𝑥 + 3 f(x) =

1

𝑥+2 f(x) =

1

𝑥−3

(h, k) form of the parent rational function f(x) = 1

𝑥

f(x) = 𝒂

𝒙−𝒉 + k

Vertical Asymptote: _______ Horizontal Asymptote: _________

When a is positive

When a is negative

1. Create a new rational function g(x) that moves the given function f(x) up 4 and left 5 units.

f(x) = 1

𝑥+3 – 9 g(x) =

2. Create a new rational function g(x) that moves the given function down 2 and right 6 units.

f(x) = 2𝑥+1

𝑥+5 g(x) =

x y

x y

x y

x y

Graph each rational function. State the domain and range. Check for any holes.

1. f(x) = 2𝑥+1

𝑥−4 2. f(x) =

−1

𝑥+2 + 1

Domain: Domain:

Range: Range:

3. f(x) = 𝑥−1

𝑥2−4𝑥+3 4.

2

2

4

4 4

xy

x x

Domain: Domain:

Range: Range:

Hole(s): Hole(s):

alg. 2 honors unit 6...

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