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2.5 – Rational Functions
Ex. 1 Graph 5
x – 2
Ex. 1 Graph 5
x – 2
Ex. 1 Graph 5
x – 2
x – 2 = 0
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (vertical asymptote)
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (asymptote)
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (asymptote)
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (asymptote)
*Graph on Calc.
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (asymptote)
*Graph on Calc.
Type: y = 5/(x – 2)
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (asymptote)
*Graph on Calc.
Type: y = 5/(x – 2)2nd Table, 3pts on each curve
Ex. 1 Graph 5
x – 2
x – 2 = 0
x = 2 (asymptote)
*Graph on Calc.
Type: y = 5/(x – 2)2nd Table, 3pts on each curve
Ex. 2 Graph x + 1
x2 + 3x + 2
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
1
x + 2
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
1
x + 2
Asymp. @ x = -2
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
1
x + 2
Asymp. @ x = -2
Hole @ x = -1
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
1
x + 2
Asymp. @ x = -2
Hole @ x = -1
Graph y = (x+1)/(x2+3x+2)
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
1
x + 2
Asymp. @ x = -2
Hole @ x = -1
*Graph y = (x+1)/(x2+3x+2)
*Then 2nd Table for 3 ordered
pairs per curve.
Ex. 2 Graph x + 1
x2 + 3x + 2
x + 1
(x + 2)(x + 1)
1
x + 2
Asymp. @ x = -2
Hole @ x = -1
*Graph y = (x+1)/(x2+3x+2)
*Then 2nd Table for 3 ordered
pairs per curve.
Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain.
f(x) = 3x2 – 3
x2 – 9
Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain.
f(x) = 3x2 – 3 = 3(x + 1)(x – 1)
x2 – 9 (x + 3)(x – 3)
Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain.
f(x) = 3x2 – 3 = 3(x + 1)(x – 1)
x2 – 9 (x + 3)(x – 3)
Vertical Asymptotes at x = -3 & x = 3
Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain.
f(x) = 3x2 – 3 = 3(x + 1)(x – 1)
x2 – 9 (x + 3)(x – 3)
Vertical Asymptotes at x = -3 & x = 3
Graph, use Table to find limits for Horizontal.
Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain.
f(x) = 3x2 – 3 = 3(x + 1)(x – 1)
x2 – 9 (x + 3)(x – 3)
Vertical Asymptotes at x = -3 & x = 3
Graph, use Table to find limits for Horizontal.
Limits show Horizontal Asymptote at y = 3.
Domain: {x | x ≠ -3, x ≠ 3}
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2)
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) =
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) = 3p
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) = 3p
2p – 4 = 3p
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) = 3p
2p – 4 = 3p
-2p -2p
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) = 3p
2p – 4 = 3p
-2p -2p
-4
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) = 3p
2p – 4 = 3p
-2p -2p
-4 =
• Proportions: 1 fraction = 1 fraction
Ex. 1 Solve each equation.
a. p _ = 2
p – 2 3
2(p – 2) = 3p
2p – 4 = 3p
-2p -2p
-4 = p
b. w + w = 4w – 3
w – 1 w – 1
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
w + w(w – 1) = 4w – 3
w + w2 – w = 4w - 3
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
w + w(w – 1) = 4w – 3
w + w2 – w = 4w – 3
w2 = 4w – 3
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
w + w(w – 1) = 4w – 3
w + w2 – w = 4w – 3
w2 = 4w – 3
w2 – 4w + 3 = 0
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
w + w(w – 1) = 4w – 3
w + w2 – w = 4w – 3
w2 = 4w – 3
w2 – 4w + 3 = 0
(w – 3)(w – 1) = 0
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
w + w(w – 1) = 4w – 3
w + w2 – w = 4w – 3
w2 = 4w – 3
w2 – 4w + 3 = 0
(w – 3)(w – 1) = 0
w = 3, w = 1
b. (w-1) w + w (w-1) = 4w – 3 (w-1)
w – 1 w – 1
w + w(w – 1) = 4w – 3
w + w2 – w = 4w – 3
w2 = 4w – 3
w2 – 4w + 3 = 0
(w – 3)(w – 1) = 0
w = 3, w = 1
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