WARM UP: Factor each completelya) 3692 xx b) xxx 28287 23 c) 168 x

d) 152 2 xx e) 8103 24 xx f) 2764 3 x

Rational Expression/ Equations:Ratio or Fraction of TWO polynomials

Previous Knowledge: (1) SIMPLIFYING

c)baa

aba33

22

a2(b – 1)

a3(1 – b)

a))7)(5(6

)7)(2(25

3

xxx

xxxb)

158

1522

2

xx

xx(x + 5)(x – 3)

(x + 5)(x + 3)

b)1610

1022

2

xx

xx(2x - 5)(x + 2)

(x + 2)(x + 8)-a3(b –1)

a2(b – 1)a

1

)5(3

)2(2

xx

x

3

3

x

x

8

52

x

x

b)2

44

67

822

2

x

x

xx

xx(x + 4)(x – 2)

(x + 6)(x + 1)

4(x + 1)

6

164

x

x

Previous Knowledge: (2) MULTIPLYING and DIVIDING

a))3(10

)2)(12(

)2)(43(

)3(44

7

xx

xx

xx

xx

c))3)(2(

)4(6

)2)(3(

)2)(4(2 72

xx

xx

xx

xxxd)

9

84

365

862

2

x

x

xx

xx

Previous Knowledge: (3) ADDITION and SUBTRACTION

4

3

)5)(4(

72

xxx

xxa) )3)(2(

7

)5)(3(

xx

x

xx

xb)

103

2

149

722

xxxx

c) 7152

5

14193

222

xx

x

xx

xd)

EVALUATING RATIONAL FUNCTIONS1. Substitute x-value into the function2. Evaluate the numerator and denominator separately3. Reduce the resulting fraction

3

7)(

x

xxf

2

103)(

2

x

xxxg

)3(f

)2(f

)3(f

)5(g

)0(g

)1(g

Practice: Evaluating Rational Functions

32

4)(

x

xxf

152

127)( 2

2

xx

xxxga) b)

)1(f

)2(f

)2(f

)4(f

)3(g

)2(g

)0(g

)4(g

Practice: Continued

xx

xxxh

6

7)( 2

23

43

9192)( 2

2

xx

xxxpc) d)

)2(p

)3(p

)1(p)0(h

)7(h

)3(h

)6(h )2(p

Number Lines and Rational Equations1) Find All Zero/ Root Values of the numerator and denominator•ZEROS in the numerator = zero in the graph (y-value)•ZEROS in the denominator = undefined in the graph (no y-value)

2) Write all the Zero/Undefined x-values on the number line

3) Between those x-values pick a number to substitute into function to determine if the graph (y-value) will be positive or negative•REASON: Only time we can change from positive to negative value is when we have a ZERO or UNDEFINED value of the graph

Example:)7)(2(

)4)(3(

xx

xx

Number Lines and Rational Equations

[1])3)(3(

)1)(5(

xx

xxx

[2])2)(3)(1(

)1)(4(

xxx

xx

Number Lines and Rational Equations

[3[932

1582

2

xx

xx

[4]xx

xxx

8

10732

23

)0(h

Unit 4: Graphing Rational EquationsGraph each of the following rational equations in your calculator and Sketch the graph on the provided axes

[1]2

3

x

xy [2] 2)3(

3

x

xy

[4]4

12

x

xy [5]

)3)(2(

1

xx

y [6]6

22

xx

xy

[3]1

12

x

xy

Investigation: Graphing Rational EquationsObservations based on the graphs #1 – 6:

(1) What is different about these graphs from previous functions that you have drawn?

Separate sections of the graph (discontinuity)Approaching specific lines (asymptotes)Holes in some graphs All decreasing, All Increasing, or Valley/ Hill Sections

(2) Is there any relationship you see between the numerator and/or denominator with the behavior of the graph?

Denominator Values Affect the vertical linesNumerator/Denominator combined affect the horizontal line

Graphing Rational Equations By Hand Basic Steps

1. Factor Numerator and Denominator

2. Determine ZERO(S) of Denominator and Numerator

3. Determine the types of Asymptotes and Discontinuity (Use zeros to help)

4. DRAW Asymptotes

5. GRAPH based on known values or positive/negative sections

#1: Vertical Asymptotes

Examples: Zeros of Denominator that do not cancel)6)(1(

4

xxVertical Asymptotes at x = 1 and x = -6

Special Behavior in Rational Equations

• x = a is a vertical asymptote if f(a) is undefined and a is a zero value of the denominator of f(x) only.

• As x approaches a from the left or right side, f(x) approaches either ±∞ “Boundary you follow along”

#1: x = a #2: x = a #3: x = a #4: x = a

#2: Points of Discontinuity (Holes in Graph)

Example: Zeros of Denominator that cancel

)5)(3(

3

152

32

xx

x

xx

x Point of Discontinuity at x = -3

Vertical Asymptote at x = 5

Special Behavior in Rational Equations

• x = a is a point of discontinuity if f(a) is undefined • a is a zero value of the numerator and denominator of f(x).• Factor (x – a) can be reduced completely from f(x)

#1: x = a#2: x = a #3: x = a

#4: x = a

Special Behavior in Rational Equations#3 Horizontal Asymptotes:

Case 1: Degree of denominator is LARGER than degree of numerator

Horizontal Asymptote: y = 0 (x – axis) 25

623

2

xx

x

135

3223

3

xx

xxCase 2: Degree of denominator is SAME AS degree of numerator

Horizontal Asymptote: y = fraction of LEADING coefficients

Case 3: Degree of denominator is SMALLER than degree of numerator

No Horizontal Asymptote: f(x) → ± ∞973

6523

24

xx

xx

• y = b is a horizontal if the end behavior of f(x) as x approaches positive or negative infinity is b.

• Note: f(x) = b on a specific domain, but is predicted not approach farther left and farther right

Example 1: Sketch two possible graphs based on each description

[A]Vertical: x = 2Horizontal: y = - 3Discontinuity: x = - 4

[2]Vertical: x = -4, x = 0Horizontal: y = 0Discontinuity: x = 2

Example 2: Determine the asymptotes and discontinuity values for the given rational equation and plot them on the given axes•FACTOR NUMERATOR AND DENOMINATOR!!!!

a)103

65)( 2

2

xx

xxxf

VA:

HA:

PD:

(x + 2)(x – 5)

(x + 2)(x + 3)

x = 5

y = 1

x = – 2

pd

Example 2 Continued

b)12

82)( 2

2

xx

xxxf

VA:

HA:

PD:

(x + 3)(x – 4)

(x + 2)(x – 4)

x = – 3

y = 1

x = 4

pd

Example 2 Continued

c)4

2)(

x

xxf

VA:

HA:

PD:

x = 4

y = 2

NA

a))3)(12)(6(

)2)(3(5)(

xxx

xxxxf

VA:

HA:

PD:

x = 6

y = 5/2

Example 3: Determine the asymptotes and discontinuity values for the given rational equation

b) 4

23 22)(

x

xxxf

VA:

HA:

PD:

x = 0

y = 0

NA

x = 6

x = 3

Example 3 Continued

c)103

)( 2

xx

xxf

VA:

HA:

PD:

(x + 2)(x – 5)

x = 5, x = – 2

y = 0

NA

d))35)(2)(14(

)2)(43)(32()(

xxx

xxxxf

VA:

HA:

PD:

x = 1/4, x = 3/5

y = 3/10

x = -2

1.

2.82

)( 2

xx

xxf

3

3)(

x

xxf

82

168)( 2

2

xx

xxxf

54

107)( 2

2

xx

xxxf

PRACTICE: Identify the vertical asymptotes, horizontal asymptotes, and points of discontinuity if they exist in each rational equation

3.

4.

5.

6. 4

1)(

2

x

xxf

82

4)(

2

xxxf

4168

)(2

xxx

xf

127

3)(

2 xx

xxf

PRACTICE: continued7.

8.