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Putting It All Together - Graphing Rational Functions

Module by: Pradnya Bhawalkar, Kim Johnston. E-mail the authors

Summary: Graphing rational functions.

When graphing rational functions, find the domain, vertical asymptotes, slant asymptotes, holes (if any), horizontal asymptotes, vertical asymptotes, zeros, and y-intercept.

To practice, graph each rational function. State the domain, hole(s), VA (vertical asyptote(s)), HA(horizontal asymptote), SA(slant asymptote), zeros, and y-intercept(y-int).

Use graph paper.

Exercise 1

r⁢x=x+1x⁢(x+4) r x x 1 x x 4

Solution

Figure 1
Figure 1 (Pit1.png)

Domain: −∞ -4 ∪ -4 0 ∪ 0 ∞ -4 -4 0 0

Hole: None

VA: x=0x0, x=-4x-4

HA: y=0y0

SA:None

Zero: (-1,0)

Y-int: None

Exercise 2

h⁢x=(2⁢x2)⁢(x−3)(x−1)⁢(x+2) h x 2 x 2 x 3 x 1 x 2

Solution

Figure 2
Figure 2 (Pit2.png)

Domain: −∞ -2 ∪ -2 1 ∪ 1 ∞ -2 -2 1 1

Hole: None

VA: x=-2x-2, x=1x1

HA:None

SA: y=2⁢x−8 y 2 x 8

Zeros: (0,0), (3,0)

Y-int: (0,0)

Exercise 3

f⁢x=3⁢x+32⁢x+4 f x 3 x 3 2 x 4

Solution

Figure 3
Figure 3 (Pit3.png)

Domain: −∞ -2 ∪ -2 ∞ -2 -2

Hole: None

VA: x=-2x-2

HA: y=32y32

SA: None

Zero: (-1,0)

Y-int: (0,3434)

Exercise 4

g⁢x=6x2−x−6 g x 6 x 2 x 6

Solution

Figure 4
Figure 4 (Pit4.png)

Domain: −∞ -2 ∪ -2 3 ∪ 3 ∞ -2 -2 3 3

Hole: None

VA: x=-2x-2, x=3x3

HA: y=0y0

SA: None

Zeros: None

Y-int: (0,-1)

Exercise 5

h⁢x=2⁢x+4x−1 h x 2 x 4 x 1

Solution

Figure 5
Figure 5 (Pit5.png)

Domain: −∞ 1 ∪ 1 ∞ 1 1

Hole: None

VA: x=1x1

HA: y=2y2

SA: None

Zero: (-2,0)

Y-int: (-2,0)

Exercise 6

t⁢x=3⁢xx2+4 t x 3 x x 2 4

Solution

Figure 6
Figure 6 (Pit6.png)

Domain: −∞ ∞

Hole: None

VA: None

HA: y=0y0

SA: None

Zero: (0,0)

Y-int: (0,0)

Exercise 7

f⁢x=x2+4x2−4 f x x 2 4 x 2 4

Solution

Figure 7
Figure 7 (Pit7.png)

Domain: −∞ -2 ∪ -2 2 ∪ 2 ∞ -2 -2 2 2

Hole: None

VA: x=-2x-2, x=2x2

HA: y=1y1

SA: None

Zeros: None

Y-int: (0,-1)

Exercise 8

f⁢x=xx+22 f x x x 2 2

Solution

Figure 8
Figure 8 (Pit8.png)

Domain: −∞ -2 ∪ -2 ∞ -2 -2

Hole: None

VA: x=-2x-2

HA: y=0y0

SA: None

Zeros: (0,0)

Y-int: (0,0)

Exercise 9

f⁢x=5⁢x2x+3 f x 5 x 2 x 3

Solution

Figure 9
Figure 9 (Pit9.png)

Domain: −∞ -3 ∪ -3 ∞ -3 -3

Hole: None

VA: x=-3x-3

HA: None

SA: y=5⁢x−15 y 5 x 15

Zeros: (0,0)

Y-int: (0,0)

Exercise 10

f⁢x=x−3x2+1 f x x 3 x 2 1

Solution

Figure 10
Figure 10 (Pit10.png)

Domain: −∞ ∞

Hole: None

VA: None

HA: y=0y0

SA:None

Zeros:(3,0)

Y-int: (0,-3)

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