General Form
The definition you must know cold
f(x) = a ยท bหฃ
where a โ 0 ยท b > 0 ยท b โ 1
a = initial amount (y-intercept, f(0) = a)
b = base or common ratio
Why b > 0?
Negative bases cause issues with fractional exponents โ e.g. (โ4)^(1/2) is undefined in the reals.
Why b โ 1?
If b = 1, then bหฃ = 1 for all x, giving f(x) = a โ just a constant, not exponential.
f(0) = a
Plug in x=0: f(0) = aยทbโฐ = aยท1 = a. So the initial value is always the y-intercept.
Growth vs. Decay
Determined entirely by the value of b (when a > 0)
๐ Exponential Growth
a > 0 and b > 1
โ Always Increasing
โช Always Concave Up
lim xโโโ = 0 ยท lim xโ+โ = โ
Example: f(x) = 3(4)หฃ
๐ Exponential Decay
a > 0 and 0 < b < 1
โ Always Decreasing
โช Always Concave Up
lim xโโโ = โ ยท lim xโ+โ = 0
Example: g(x) = 5(2/3)หฃ
Key Properties
What makes exponentials special
| Property | Growth (a>0, b>1) | Decay (a>0, 0<b<1) | Reflected (a<0, b>1) |
|---|---|---|---|
| Always Increasing | โ | โ | โ |
| Always Decreasing | โ | โ | โ |
| Concave Up | โ | โ | โ |
| Concave Down | โ | โ | โ |
| Local Extrema | โ Never โ no relative max or min ever | ||
| Points of Inflection | โ Never โ concavity never changes | ||
| Horizontal Asymptote | y = 0 (the x-axis), unless the function is vertically shifted | ||
End Behavior
Writing limit statements โ Example 1
The three possible end behaviors
As x โ ยฑโ, the output of abหฃ will do one of three things: approach โ, โโ, or 0.
Key questions: Is a positive or negative? Is b greater or less than 1? Which end are you looking at?
๐ Example 1a โ Growth graph: a > 0, b > 1
L
x โ โโ: the exponential shrinks toward 0 from above (horizontal asymptote)R
x โ +โ: the exponential grows without boundlim(xโโโ) abหฃ = 0 lim(xโ+โ) abหฃ = โ
๐ Example 1b โ Reflected graph: a < 0, b > 1 (graph is below x-axis, falls to the right)
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x โ โโ: large positive exponent ร negative a โ falls to โโR
x โ +โ: the negative graph approaches 0 from belowlim(xโโโ) abหฃ = โโ lim(xโ+โ) abหฃ = 0
๐ Example 1c โ g(x) = 5(2/3)หฃ โ decay since b = 2/3, so 0 < b < 1
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b = 2/3 < 1 โ this is decay (decreasing, approaches 0 on the right)L
x โ โโ: (2/3) raised to large negative power = large positive โ โR
x โ +โ: fraction raised to large positive power โ 0lim(xโโโ) g(x) = โ lim(xโ+โ) g(x) = 0
Increasing, Decreasing & Concavity
Reading from the formula โ Example 2
a > 0, b > 1
Increasing and Concave Up
e.g. h(x) = 3(4)หฃ โ b=4>1 โ
a > 0, 0 < b < 1
Decreasing and Concave Up
e.g. g(x) = 5(2/3)หฃ โ b=2/3 โ
a < 0, b > 1
Decreasing and Concave Down
Reflected growth โ graph below x-axis
a < 0, 0 < b < 1
Increasing and Concave Down
Reflected decay โ rises to the right
๐ Example 2c โ h(x) = 3(4)หฃ. Is it increasing or decreasing? Concave up or down?
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Identify: a = 3 > 0 and b = 4 > 1 โ exponential growth2
Growth with positive a โ always increasing, always concave upIncreasing ยท Concave Up
Identifying from a Table
Example 3 โ the vertical translation trick
The strategy
Step 1: Check if differences are constant โ linear.
Step 2: Check if ratios are constant โ exponential.
Step 3: If neither โ try subtracting a constant c from all f(x) values and check ratios again. If that works, f(x) is a vertical translation of an exponential.
๐ Example 3 โ Is f linear, quadratic, exponential, or none of these?
| x | 1 | 6 | 11 | 16 | 21 |
|---|---|---|---|---|---|
| f(x) | 3 | 5 | 9 | 17 | 33 |
| g(x) = f(x) โ 1 | 2 | 4 | 8 | 16 | 32 |
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Differences of f(x): 2, 4, 8, 16 โ not constant โ not linear2
Ratios of f(x): 5/3 โ 1.67, 9/5 = 1.8 โ not constant โ not purely exponential3
Try g(x) = f(x) โ 1: values are 2, 4, 8, 16, 32 = 2ยน, 2ยฒ, 2ยณ, 2โด, 2โต4
Ratios of g(x): 4/2=2, 8/4=2, 16/8=2, 32/16=2 โ constant ratio r = 2 โ5
So g(x) is exponential, meaning f(x) = g(x) + 1 is a vertical translation of an exponentialf(x) is a vertical translation of an exponential function
Quick Reference
Screenshot and save this!
๐ The Formula
f(x) = a ยท bหฃ
b > 0, b โ 1, a โ 0
a = f(0)
Initial value = y-intercept
b = f(n+1) / f(n)
Common ratio from any two consecutive outputs
๐ End Behavior Quick Guide
a>0, b>1 โ 0 left, โ right
Growth โ rises right
a>0, 0<b<1 โ โ left, 0 right
Decay โ rises left
a<0, b>1 โ โโ left, 0 right
Reflected growth โ falls right
โ ๏ธ Common Mistakes
โ Thinking b can be negative
b must be positive (b > 0). A negative base causes undefined values for fractional exponents.
โ Confusing growth/decay when a < 0
If a < 0 and b > 1, the function is actually decreasing and concave down โ not growth!
โ Saying exponentials have local extrema
Exponentials are always strictly increasing or always decreasing โ they NEVER have local max or min.
โ Giving up if ratios aren't constant in a table
Try subtracting a constant first โ you may have a vertical translation of an exponential.