H Algebra 2/Trig · Unit 7
Sketch y = a·bx + c, name growth or decay, and list the asymptote, domain, range, and end behavior.
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The parent graph
y = bx passes through (0, 1). Multiply by a, then move up or down by c.
y = 2x is growth. It passes through (0, 1), and the asymptote is y = 0.
Asymptote, domain, and range
For y = a·bx + c, the horizontal asymptote is y = c.
The domain is all real numbers. The range is above y = c when a is positive, and below y = c when a is negative.
Growth or decay
b greater than 1 is growth. b between 0 and 1 is decay.
A negative a reflects the graph over the asymptote. It is still growth or decay.
End behavior
One end runs away from the asymptote. The other end gets closer and closer to y = c and never touches it.
Positive a, no horizontal shift
f(x) = 8(12)x − 3
x = −1
f(−1) = 8(12)⁻¹ − 3
(12)⁻¹ = 2
8 · 2 = 16
16 − 3 = 13
f(−1) = 13
(−1, 13)
f(x) = 8(12)x − 3
x = 0
f(0) = 8(12)⁰ − 3
(12)⁰ = 1
8 · 1 = 8
8 − 3 = 5
f(0) = 5
(0, 5)
f(x) = 8(12)x − 3
x = 1
f(1) = 8(12)¹ − 3
(12)¹ = 12
8 · 12 = 4
4 − 3 = 1
f(1) = 1
(1, 1)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 12
Decay
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = −3
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 8, c = −3
(−3, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −3
As x → −∞, y → ∞
Answer: Decay; asymptote y = −3; domain (−∞, ∞); range (−3, ∞); as x → ∞, y → −3; as x → −∞, y → ∞
f(x) = 2(2x) − 6
x = −1
f(−1) = 2(2⁻¹) − 6
2⁻¹ = 12
2 · 12 = 1
1 − 6 = −5
f(−1) = −5
(−1, −5)
f(x) = 2(2x) − 6
x = 0
f(0) = 2(2⁰) − 6
2⁰ = 1
2 · 1 = 2
2 − 6 = −4
f(0) = −4
(0, −4)
f(x) = 2(2x) − 6
x = 1
f(1) = 2(2¹) − 6
2¹ = 2
2 · 2 = 4
4 − 6 = −2
f(1) = −2
(1, −2)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 2
Growth
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = −6
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 2, c = −6
(−6, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → ∞
As x → −∞, y → −6
Answer: Growth; asymptote y = −6; domain (−∞, ∞); range (−6, ∞); as x → ∞, y → ∞; as x → −∞, y → −6
f(x) = 4(12)x + 1
x = −1
f(−1) = 4(12)⁻¹ + 1
(12)⁻¹ = 2
4 · 2 = 8
8 + 1 = 9
f(−1) = 9
(−1, 9)
f(x) = 4(12)x + 1
x = 0
f(0) = 4(12)⁰ + 1
(12)⁰ = 1
4 · 1 = 4
4 + 1 = 5
f(0) = 5
(0, 5)
f(x) = 4(12)x + 1
x = 1
f(1) = 4(12)¹ + 1
(12)¹ = 12
4 · 12 = 2
2 + 1 = 3
f(1) = 3
(1, 3)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 12
Decay
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 1
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 4, c = 1
(1, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → 1
As x → −∞, y → ∞
Answer: Decay; asymptote y = 1; domain (−∞, ∞); range (1, ∞); as x → ∞, y → 1; as x → −∞, y → ∞
f(x) = 3(3x) + 1
x = −1
f(−1) = 3(3⁻¹) + 1
3⁻¹ = 13
3 · 13 = 1
1 + 1 = 2
f(−1) = 2
(−1, 2)
f(x) = 3(3x) + 1
x = 0
f(0) = 3(3⁰) + 1
3⁰ = 1
3 · 1 = 3
3 + 1 = 4
f(0) = 4
(0, 4)
f(x) = 3(3x) + 1
x = 1
f(1) = 3(3¹) + 1
3¹ = 3
3 · 3 = 9
9 + 1 = 10
f(1) = 10
(1, 10)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 3
Growth
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 1
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 3, c = 1
(1, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → ∞
As x → −∞, y → 1
Answer: Growth; asymptote y = 1; domain (−∞, ∞); range (1, ∞); as x → ∞, y → ∞; as x → −∞, y → 1
f(x) = 6(13)x − 2
x = −1
f(−1) = 6(13)⁻¹ − 2
(13)⁻¹ = 3
6 · 3 = 18
18 − 2 = 16
f(−1) = 16
(−1, 16)
f(x) = 6(13)x − 2
x = 0
f(0) = 6(13)⁰ − 2
(13)⁰ = 1
6 · 1 = 6
6 − 2 = 4
f(0) = 4
(0, 4)
f(x) = 6(13)x − 2
x = 1
f(1) = 6(13)¹ − 2
(13)¹ = 13
6 · 13 = 2
2 − 2 = 0
f(1) = 0
(1, 0)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 13
Decay
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = −2
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 6, c = −2
(−2, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −2
As x → −∞, y → ∞
Answer: Decay; asymptote y = −2; domain (−∞, ∞); range (−2, ∞); as x → ∞, y → −2; as x → −∞, y → ∞
f(x) = 4(2x) + 2
x = −1
f(−1) = 4(2⁻¹) + 2
2⁻¹ = 12
4 · 12 = 2
2 + 2 = 4
f(−1) = 4
(−1, 4)
f(x) = 4(2x) + 2
x = 0
f(0) = 4(2⁰) + 2
2⁰ = 1
4 · 1 = 4
4 + 2 = 6
f(0) = 6
(0, 6)
f(x) = 4(2x) + 2
x = 1
f(1) = 4(2¹) + 2
2¹ = 2
4 · 2 = 8
8 + 2 = 10
f(1) = 10
(1, 10)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 2
Growth
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 2
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 4, c = 2
(2, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → ∞
As x → −∞, y → 2
Answer: Growth; asymptote y = 2; domain (−∞, ∞); range (2, ∞); as x → ∞, y → ∞; as x → −∞, y → 2
Negative a, reflected over the asymptote
f(x) = −2(2x) + 5
x = −1
f(−1) = −2(2⁻¹) + 5
2⁻¹ = 12
−2 · 12 = −1
−1 + 5 = 4
f(−1) = 4
(−1, 4)
f(x) = −2(2x) + 5
x = 0
f(0) = −2(2⁰) + 5
2⁰ = 1
−2 · 1 = −2
−2 + 5 = 3
f(0) = 3
(0, 3)
f(x) = −2(2x) + 5
x = 1
f(1) = −2(2¹) + 5
2¹ = 2
−2 · 2 = −4
−4 + 5 = 1
f(1) = 1
(1, 1)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 2
Growth, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 5
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −2, c = 5
(−∞, 5)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −∞
As x → −∞, y → 5
Answer: Growth, reflected over the asymptote; asymptote y = 5; domain (−∞, ∞); range (−∞, 5); as x → ∞, y → −∞; as x → −∞, y → 5
f(x) = −2(2x) + 6
x = −1
f(−1) = −2(2⁻¹) + 6
2⁻¹ = 12
−2 · 12 = −1
−1 + 6 = 5
f(−1) = 5
(−1, 5)
f(x) = −2(2x) + 6
x = 0
f(0) = −2(2⁰) + 6
2⁰ = 1
−2 · 1 = −2
−2 + 6 = 4
f(0) = 4
(0, 4)
f(x) = −2(2x) + 6
x = 1
f(1) = −2(2¹) + 6
2¹ = 2
−2 · 2 = −4
−4 + 6 = 2
f(1) = 2
(1, 2)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 2
Growth, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 6
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −2, c = 6
(−∞, 6)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −∞
As x → −∞, y → 6
Answer: Growth, reflected over the asymptote; asymptote y = 6; domain (−∞, ∞); range (−∞, 6); as x → ∞, y → −∞; as x → −∞, y → 6
f(x) = −4(12)x + 1
x = −1
f(−1) = −4(12)⁻¹ + 1
(12)⁻¹ = 2
−4 · 2 = −8
−8 + 1 = −7
f(−1) = −7
(−1, −7)
f(x) = −4(12)x + 1
x = 0
f(0) = −4(12)⁰ + 1
(12)⁰ = 1
−4 · 1 = −4
−4 + 1 = −3
f(0) = −3
(0, −3)
f(x) = −4(12)x + 1
x = 1
f(1) = −4(12)¹ + 1
(12)¹ = 12
−4 · 12 = −2
−2 + 1 = −1
f(1) = −1
(1, −1)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 12
Decay, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 1
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −4, c = 1
(−∞, 1)
End behavior depends on the base and on the sign of a.
As x → ∞, y → 1
As x → −∞, y → −∞
Answer: Decay, reflected over the asymptote; asymptote y = 1; domain (−∞, ∞); range (−∞, 1); as x → ∞, y → 1; as x → −∞, y → −∞
f(x) = −3(3x) − 2
x = −1
f(−1) = −3(3⁻¹) − 2
3⁻¹ = 13
−3 · 13 = −1
−1 − 2 = −3
f(−1) = −3
(−1, −3)
f(x) = −3(3x) − 2
x = 0
f(0) = −3(3⁰) − 2
3⁰ = 1
−3 · 1 = −3
−3 − 2 = −5
f(0) = −5
(0, −5)
f(x) = −3(3x) − 2
x = 1
f(1) = −3(3¹) − 2
3¹ = 3
−3 · 3 = −9
−9 − 2 = −11
f(1) = −11
(1, −11)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 3
Growth, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = −2
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −3, c = −2
(−∞, −2)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −∞
As x → −∞, y → −2
Answer: Growth, reflected over the asymptote; asymptote y = −2; domain (−∞, ∞); range (−∞, −2); as x → ∞, y → −∞; as x → −∞, y → −2
f(x) = −6(13)x + 4
x = −1
f(−1) = −6(13)⁻¹ + 4
(13)⁻¹ = 3
−6 · 3 = −18
−18 + 4 = −14
f(−1) = −14
(−1, −14)
f(x) = −6(13)x + 4
x = 0
f(0) = −6(13)⁰ + 4
(13)⁰ = 1
−6 · 1 = −6
−6 + 4 = −2
f(0) = −2
(0, −2)
f(x) = −6(13)x + 4
x = 1
f(1) = −6(13)¹ + 4
(13)¹ = 13
−6 · 13 = −2
−2 + 4 = 2
f(1) = 2
(1, 2)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 13
Decay, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 4
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −6, c = 4
(−∞, 4)
End behavior depends on the base and on the sign of a.
As x → ∞, y → 4
As x → −∞, y → −∞
Answer: Decay, reflected over the asymptote; asymptote y = 4; domain (−∞, ∞); range (−∞, 4); as x → ∞, y → 4; as x → −∞, y → −∞
f(x) = −8(12)x + 5
x = −1
f(−1) = −8(12)⁻¹ + 5
(12)⁻¹ = 2
−8 · 2 = −16
−16 + 5 = −11
f(−1) = −11
(−1, −11)
f(x) = −8(12)x + 5
x = 0
f(0) = −8(12)⁰ + 5
(12)⁰ = 1
−8 · 1 = −8
−8 + 5 = −3
f(0) = −3
(0, −3)
f(x) = −8(12)x + 5
x = 1
f(1) = −8(12)¹ + 5
(12)¹ = 12
−8 · 12 = −4
−4 + 5 = 1
f(1) = 1
(1, 1)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 12
Decay, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 5
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −8, c = 5
(−∞, 5)
End behavior depends on the base and on the sign of a.
As x → ∞, y → 5
As x → −∞, y → −∞
Answer: Decay, reflected over the asymptote; asymptote y = 5; domain (−∞, ∞); range (−∞, 5); as x → ∞, y → 5; as x → −∞, y → −∞
A horizontal shift inside the exponent
f(x) = 3(13)x − 2 + 1
x = 1
f(1) = 3(13)⁻¹ + 1
(13)⁻¹ = 3
3 · 3 = 9
9 + 1 = 10
f(1) = 10
(1, 10)
f(x) = 3(13)x − 2 + 1
x = 2
f(2) = 3(13)⁰ + 1
(13)⁰ = 1
3 · 1 = 3
3 + 1 = 4
f(2) = 4
(2, 4)
f(x) = 3(13)x − 2 + 1
x = 3
f(3) = 3(13)¹ + 1
(13)¹ = 13
3 · 13 = 1
1 + 1 = 2
f(3) = 2
(3, 2)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 13
Decay
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 1
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 3, c = 1
(1, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → 1
As x → −∞, y → ∞
Answer: Decay; asymptote y = 1; domain (−∞, ∞); range (1, ∞); as x → ∞, y → 1; as x → −∞, y → ∞
f(x) = 4(2x − 3) − 1
x = 2
f(2) = 4(2⁻¹) − 1
2⁻¹ = 12
4 · 12 = 2
2 − 1 = 1
f(2) = 1
(2, 1)
f(x) = 4(2x − 3) − 1
x = 3
f(3) = 4(2⁰) − 1
2⁰ = 1
4 · 1 = 4
4 − 1 = 3
f(3) = 3
(3, 3)
f(x) = 4(2x − 3) − 1
x = 4
f(4) = 4(2¹) − 1
2¹ = 2
4 · 2 = 8
8 − 1 = 7
f(4) = 7
(4, 7)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 2
Growth
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = −1
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 4, c = −1
(−1, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → ∞
As x → −∞, y → −1
Answer: Growth; asymptote y = −1; domain (−∞, ∞); range (−1, ∞); as x → ∞, y → ∞; as x → −∞, y → −1
f(x) = 6(12)x + 1 + 2
x = −2
f(−2) = 6(12)⁻¹ + 2
(12)⁻¹ = 2
6 · 2 = 12
12 + 2 = 14
f(−2) = 14
(−2, 14)
f(x) = 6(12)x + 1 + 2
x = −1
f(−1) = 6(12)⁰ + 2
(12)⁰ = 1
6 · 1 = 6
6 + 2 = 8
f(−1) = 8
(−1, 8)
f(x) = 6(12)x + 1 + 2
x = 0
f(0) = 6(12)¹ + 2
(12)¹ = 12
6 · 12 = 3
3 + 2 = 5
f(0) = 5
(0, 5)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 12
Decay
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 2
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 6, c = 2
(2, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → 2
As x → −∞, y → ∞
Answer: Decay; asymptote y = 2; domain (−∞, ∞); range (2, ∞); as x → ∞, y → 2; as x → −∞, y → ∞
f(x) = −3(3x + 1) + 6
x = −2
f(−2) = −3(3⁻¹) + 6
3⁻¹ = 13
−3 · 13 = −1
−1 + 6 = 5
f(−2) = 5
(−2, 5)
f(x) = −3(3x + 1) + 6
x = −1
f(−1) = −3(3⁰) + 6
3⁰ = 1
−3 · 1 = −3
−3 + 6 = 3
f(−1) = 3
(−1, 3)
f(x) = −3(3x + 1) + 6
x = 0
f(0) = −3(3¹) + 6
3¹ = 3
−3 · 3 = −9
−9 + 6 = −3
f(0) = −3
(0, −3)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 3
Growth, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 6
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −3, c = 6
(−∞, 6)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −∞
As x → −∞, y → 6
Answer: Growth, reflected over the asymptote; asymptote y = 6; domain (−∞, ∞); range (−∞, 6); as x → ∞, y → −∞; as x → −∞, y → 6
f(x) = 5(15)x − 1 − 2
x = 0
f(0) = 5(15)⁻¹ − 2
(15)⁻¹ = 5
5 · 5 = 25
25 − 2 = 23
f(0) = 23
(0, 23)
f(x) = 5(15)x − 1 − 2
x = 1
f(1) = 5(15)⁰ − 2
(15)⁰ = 1
5 · 1 = 5
5 − 2 = 3
f(1) = 3
(1, 3)
f(x) = 5(15)x − 1 − 2
x = 2
f(2) = 5(15)¹ − 2
(15)¹ = 15
5 · 15 = 1
1 − 2 = −1
f(2) = −1
(2, −1)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 15
Decay
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = −2
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = 5, c = −2
(−2, ∞)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −2
As x → −∞, y → ∞
Answer: Decay; asymptote y = −2; domain (−∞, ∞); range (−2, ∞); as x → ∞, y → −2; as x → −∞, y → ∞
f(x) = −4(2x + 2) + 3
x = −3
f(−3) = −4(2⁻¹) + 3
2⁻¹ = 12
−4 · 12 = −2
−2 + 3 = 1
f(−3) = 1
(−3, 1)
f(x) = −4(2x + 2) + 3
x = −2
f(−2) = −4(2⁰) + 3
2⁰ = 1
−4 · 1 = −4
−4 + 3 = −1
f(−2) = −1
(−2, −1)
f(x) = −4(2x + 2) + 3
x = −1
f(−1) = −4(2¹) + 3
2¹ = 2
−4 · 2 = −8
−8 + 3 = −5
f(−1) = −5
(−1, −5)
Growth or decay: b greater than 1 means growth. b between 0 and 1 means decay.
b = 2
Growth, reflected over the asymptote
Horizontal asymptote of y = a·b^(x − h) + c:
y = c
y = 3
Domain of an exponential function:
all real numbers
(−∞, ∞)
Range: above the asymptote when a is positive, and below it when a is negative.
a = −4, c = 3
(−∞, 3)
End behavior depends on the base and on the sign of a.
As x → ∞, y → −∞
As x → −∞, y → 3
Answer: Growth, reflected over the asymptote; asymptote y = 3; domain (−∞, ∞); range (−∞, 3); as x → ∞, y → −∞; as x → −∞, y → 3
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