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Topic 2.3 ยท General Form
What is the general form of an exponential function? State all conditions.
f(x) = a ยท bหฃ
What must be true about a and b?
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Topic 2.3 ยท Growth
What conditions on a and b produce exponential growth?
Growth means the function increases as x increases
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Topic 2.3 ยท Decay
What conditions on a and b produce exponential decay?
Decay means the function decreases as x increases
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Topic 2.3 ยท End Behavior
For f(x) = aยทbหฃ with a > 0 and b > 1 (growth), write the limit statements.
What happens as x โ +โ and x โ โโ?
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Topic 2.3 ยท End Behavior
For g(x) = 5(2/3)หฃ, write both limit statements.
g(x) = 5(2/3)หฃ
Is this growth or decay? What is b?
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Topic 2.3 ยท Concavity
h(x) = 3(4)หฃ โ is it increasing or decreasing? Concave up or down?
h(x) = 3(4)หฃ
Check b for direction, check a for concavity
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Topic 2.3 ยท No Extrema
Can an exponential function f(x) = aยทbหฃ have a local maximum or minimum?
Think about whether it can ever change direction
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Topic 2.3 ยท Table Identification
Table shows f(x) = 3, 5, 9, 17, 33 at equally spaced x. What type of function is f?
g(x) = f(x) โ 1 gives 2, 4, 8, 16, 32
Check g(x) ratios
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โ Answer โ Growth Conditions
a > 0 and b > 1
Both conditions required. Function is always increasing, concave up, approaches 0 left and +โ right.
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โ Answer โ General Form
f(x) = aยทbหฃ, b > 0, b โ 1, a โ 0
b > 0 avoids undefined values. b โ 1 avoids a constant function. a = f(0) = y-intercept.
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โ Answer โ Growth Limits
lim xโโโ = 0 ยท lim xโ+โ = +โ
As xโโโ: bหฃโ0 so fโ0.
As xโ+โ: bหฃโโ so fโ+โ.
The x-axis (y=0) is a horizontal asymptote on the left.
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โ Answer โ Decay Conditions
a > 0 and 0 < b < 1
The base is a fraction between 0 and 1. Function is always decreasing, still concave up (a > 0), approaches +โ left and 0 right.
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โ Answer โ h(x) = 3(4)หฃ
Increasing ยท Concave Up
b = 4 > 1 โ increasing.
a = 3 > 0 โ concave up.
No local extrema. No inflection points.
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โ Answer โ g(x) = 5(2/3)หฃ Limits
lim xโโโ = +โ ยท lim xโ+โ = 0
b = 2/3, so 0 < b < 1 โ decay.
As xโ+โ: (2/3)หฃโ0. As xโโโ: (2/3)หฃโ+โ.
a = 5 > 0 keeps it positive.
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โ Answer โ Table Identification
Vertical translation of an exponential
g(x) = f(x)โ1 gives 2,4,8,16,32 โ ratio = 2 โ.
So f(x) = g(x)+1 is an exponential + constant.
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โ Answer โ No Local Extrema
No โ never any local extrema
Exponentials are always increasing or always decreasing โ they never switch direction. Also no inflection points โ concavity never changes.