An exponential function is a function in which the variable, x, appears in the exponent, rather than in the base. The general form of an exponential function is , where a is the initial value, also known as the y-intercept, and b is the base, which is a positive number other than 1.
The behavior of an exponential function depends on the value of the base, b. When the base, b, is greater than 1 (b > 1), the exponential function demonstrates exponential growth. This means that as x increases, the value of the function, f(x), increases at an increasingly rapid rate. The larger the base, the faster the growth.
On the other hand, when the base, b, is between 0 and 1 (0 < b < 1), the exponential function demonstrates exponential decay. This means that as x increases, the value of the function, f(x), decreases at an increasingly rapid rate. The smaller the base, the faster the decay.
It's important to note that in both cases the initial value a must be greater than 0, otherwise the function would not be defined.
Domain
The domain of an exponential function is all real numbers, which means that the function can be evaluated for any real number value of x. 🫂
When the natural numbers (1, 2, 3, 4, ...) are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function's initial value. For example, when x = 1, the function value is , which is equal to ab. When x = 2, the function value is , which is equal to . In general, when x = n, the function value is , which is equal to , meaning that the base b is being multiplied by itself n times.
This idea can be used to model real-world situations that involve exponential growth or decay, such as compound interest, population growth, and radioactive decay. For example, if a population is growing at a rate of 5% per year, the population after n years will be , where P is the initial population. Here the base 1.05 represents the growth rate of 5% and the exponent represents the number of years. 🌆
↕️ Trends: Increasing vs. Decreasing, Concave Up vs. Down
Because the output values of exponential functions are proportional over equal-length input-value intervals, the graphs of exponential functions are always increasing or always decreasing.
Depending on the value of the base, b, the function will have an upward or downward concavity. If the base, b, is greater than 1, the exponential function demonstrates exponential growth, and its graph is concave up. If the base, b, is between 0 and 1, the exponential function demonstrates exponential decay, and its graph is concave down.
Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have inflection points. The reason is that an inflection point occurs when a graph changes from concave up to concave down or vice versa, but since exponential functions are always concave up or always concave down, they do not have inflection points.
An extrema is the highest or lowest value of a function, but exponential functions are always increasing or always decreasing and therefore, don't have extrema on an open interval.
Additive Transformations
An additive transformation of a function f(x) is a function of the form , where k is a constant. The additive transformation function g(x) shifts the graph of f(x) vertically by k units.
If the values of the additive transformation function of any function f are proportional over equal length input-value intervals, then f is exponential. This can be proven by observing that if the values of g(x) are proportional over equal length input-value intervals, then the graph of g(x) is exponential, and since , the graph of f(x) is also exponential.
Exponential functions are functions that change proportionally over equal-length input-value intervals. This is, the ratio of the y-coordinates of two points on the graph of the function is the same for any two points. In other words, the graph of an exponential function is always increasing or always decreasing, and it's concave up if the base is greater than 1 or concave down if the base is between 0 and 1. 🫡
If we add a constant k to the function f(x) the graph will be shifted k units up or down but the shape of the graph will not change, this means that if the graph of is an exponential function, then f(x) is also an exponential function.
🛑 Limits
For an exponential function in general form, as the input values increase without bound, the output values will increase without bound or will get arbitrarily close to zero. If the base of the exponential function, b, is greater than 1, the output values of the function will increase without bound as x approaches positive infinity. The limit of the function as x approaches positive infinity is ∞. This means that as x gets larger and larger, the value of the function gets larger and larger without bound. 🪐
On the other hand, if the base of the exponential function, b, is between 0 and 1, the output values of the function will decrease without bound as x approaches positive infinity. The limit of the function as x approaches positive infinity is 0. This means that as x gets larger and larger, the value of the function gets closer and closer to zero.
The same behavior occurs when x approaches negative infinity. If the base of the exponential function, b, is greater than 1, the output values of the function will decrease without bound as x approaches negative infinity. The limit of the function as x approaches negative infinity is -∞. This means that as x gets more negative, the value of the function gets more negative without bound. 🔻
And if the base of the exponential function, b, is between 0 and 1, the output values of the function will increase without bound as x approaches negative infinity. The limit of the function as x approaches negative infinity is 0. This means that as x gets more negative, the value of the function gets closer and closer to zero.
Overall, this means that we have three possible scenarios for an exponential function in general form:
Frequently Asked Questions
How do I tell if a function is exponential growth or decay?
Look at the function in the form f(x) = a·b^x (a ≠ 0, b > 0, b ≠ 1). The sign and size of a and b tell you growth vs. decay: - If a > 0 and b > 1 → exponential growth (values increase as x increases). - If a > 0 and 0 < b < 1 → exponential decay (values decrease as x increases). You can also tell from data/graphs: equal-length x-intervals produce proportional (multiplicative) changes—constant ratio per interval. Graphically, growth is always increasing (no extrema), decay is always decreasing; both have horizontal asymptote y = 0 and domain (−∞, ∞). For modeling or exam work, be ready to rewrite forms (e.g., continuous compounding with e) and use the initial value a as the y-intercept. Topic study guide and practice sets can help you spot these quickly (study guide: https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy; unit overview: https://library.fiveable.me/ap-pre-calculus/unit-2; practice: https://library.fiveable.me/practice/ap-pre-calculus).
What's the difference between f(x) = ab^x and f(x) = a + bx?
They look similar but are totally different types of functions. - Form/meaning: f(x)=a b^x is exponential (initial value a, base b>0, b≠1). Each equal step in x multiplies the output by b (multiplicative change). f(x)=a+bx is linear: a is the y-intercept and b is the constant slope (additive change—you add b for each unit increase in x). - Key features (from the CED): Exponential has domain all real numbers and range positive reals (if a>0), a horizontal asymptote y=0, and is always increasing (b>1) or always decreasing (0
I'm confused about when b > 1 vs when 0 < b < 1 in exponential functions - can someone explain?
Think of f(x) = a b^x as starting at a (the y-intercept) and multiplying by b for every +1 in x. If a > 0: - b > 1 → exponential growth. Each 1-step right multiplies the output by a factor >1, so values get larger as x increases. The graph is increasing, always concave up, has no extrema, domain = all real numbers, range = positive reals, and a horizontal asymptote y = 0 as x → −∞. (Example: b = 2 doubles each step.) - 0 < b < 1 → exponential decay. Each 1-step right multiplies by a factor <1, so values shrink as x increases. The graph is decreasing, always concave down, same domain/range/asymptote behavior (approaches 0 as x → ∞). (Example: b = 1/2 halves each step; half-life/doubling time ideas follow.) If a < 0, the sign flips but monotonicity and asymptotic behavior are similar (outputs negative). For more AP-aligned examples and practice, see the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and lots of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
What does it mean that exponential functions are proportional over equal intervals?
It means that if you pick any equal-length step in x, the outputs change by the same multiplicative factor. For f(x)=a·b^x and any fixed h, f(x+h)/f(x)=b^h is constant for all x—so over every interval of length h the outputs are proportional (same ratio). That’s why exponentials have constant percentage (not constant additive) change, are either always increasing (b>1) or always decreasing (0
How do I find the initial value and base of an exponential function from a graph?
Read the graph and pick clear points. The initial value a = f(0) is the y-intercept (the height where the graph crosses x = 0). The base b tells how outputs scale for a 1-unit input change. If you can read f(0)=a and f(1), then b = f(1)/a because f(1)=ab. If you don’t have x = 1, use two points (x1, y1) and (x2, y2) with y1,y2 ≠ 0: b = (y2 / y1)^(1/(x2−x1)) and a = y1 / b^(x1). Check: b>1 means growth, 0
When do I use exponential functions vs linear functions in word problems?
Use linear when the quantity changes by the same amount each equal time step (constant additive change). Use exponential when the quantity changes by the same factor (percent) over equal intervals (constant multiplicative change). Quick checks: - If differences between successive values are constant → model with y = mx + b (linear). - If ratios between successive values are (roughly) constant → model with f(x) = a·b^x (exponential). Examples: saving $50 every month → linear; money growing by 6.1% each quarter or half-life problems → exponential (CED 2.3.A.1, keywords: exponential growth/decay, initial value a, base b, doubling time/half-life). Graph/clue: exponential graphs are always increasing or decreasing and have horizontal asymptote y=0; linear graphs have constant slope and no asymptote. For data, check residuals or do regressions (AP exam may ask modeling and residual interpretation; Topic 2.3 study guide covers this—(https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy)). For more practice, use unit review (https://library.fiveable.me/ap-pre-calculus/unit-2) and the practice bank (https://library.fiveable.me/practice/ap-pre-calculus).
Why don't exponential functions have maximum or minimum points?
Because exponential functions in the form f(x) = a b^x (a ≠ 0, b > 0, b ≠ 1) are either always increasing (b > 1) or always decreasing (0 < b < 1), they don’t have interior maxima or minima. The CED calls this monotonicity: outputs change proportionally over equal x-intervals, so the graph never turns around—no local extrema and no inflection points (2.3.A.1–2.3.A.3). Their domain is all real numbers, so there’s no endpoint to force an extremum unless you restrict to a closed interval. Also, for a > 0 the range is (0, ∞) and the graph approaches a horizontal asymptote (often y = 0) as x → −∞ or +∞, so it just gets arbitrarily close to a value instead of reaching a max or min. For practice seeing these ideas on graphs and AP-style problems, check the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and Unit 2 overview (https://library.fiveable.me/ap-pre-calculus/unit-2).
What's the formula for exponential growth and decay problems?
The basic formula is f(x) = a b^x, where a ≠ 0 is the initial value (y-intercept) and b > 0, b ≠ 1 is the base. - If a > 0 and b > 1 → exponential growth. - If a > 0 and 0 < b < 1 → exponential decay. Common equivalent forms you’ll see on the AP: - Discrete/periodic growth/decay: A(t) = A0 · b^t (t in number of periods). - Continuous growth/decay (natural base e): A(t) = A0 · e^{kt}, where k>0 grows, k<0 decays. Useful derived formulas: - Half-life (decay): A(t)=A0·(1/2)^{t/T½} or T½ = ln(1/2)/k for A0·e^{kt}. - Doubling time (growth): T_double = ln(2)/k for A0·e^{kt}. Key AP facts: domain = all real numbers, range = positive reals, horizontal asymptote y=0, monotonic and no inflection points. For more practice and examples, see the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and Unit 2 overview (https://library.fiveable.me/ap-pre-calculus/unit-2).
I don't understand why exponential functions are always concave up or down - how do I tell which?
Think of f(x) = a b^x. Its shape comes from the second derivative. Compute quickly: f′(x) = a b^x ln b f″(x) = a b^x (ln b)^2 Since b>0 and b≠1, (ln b)^2 is always positive. Also b^x>0 for all x. So the sign of f″(x) is the sign of a for every x. That means: - If a > 0, f″(x) > 0 for all x ⇒ graph is always concave up. - If a < 0, f″(x) < 0 for all x ⇒ graph is always concave down. Because f″ never changes sign, exponential graphs have no inflection points and no interior extrema (they’re always increasing or decreasing depending on b and a). Example: f(x)=2·3^x (a=2) is concave up; g(x)=−1·(1/2)^x (a=−1) is concave down. This matches the CED: exponentials are monotonic and always concave (2.3.A.1–3). For more explanation and practice, check the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and AP Precalc practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
How do I solve for x in equations like 3^x = 27?
Solve it by rewriting both sides with the same base or using logs. 1) Same-base method (best when possible): 27 = 3^3, so 3^x = 3^3 ⇒ x = 3. That's the exact value—always prefer exact forms on AP free-response. 2) Log method (works always): take natural logs (or any log) of both sides: x ln 3 = ln 27 ⇒ x = ln 27 / ln 3. That simplifies to x = 3 because ln 27 = ln(3^3) = 3 ln 3. Note: Exponential functions are of the form a b^x (CED 2.3.A.1). Their domain is all real numbers and outputs are positive, so solving often uses rewriting bases or log rules (CED 2.3.A.2, 2.3.A.5). For more practice and examples, check the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and the AP Precalculus practice set (https://library.fiveable.me/practice/ap-pre-calculus).
What happens to exponential functions as x approaches infinity?
As x → ∞, exponential functions f(x) = a·b^x behave in one of a few predictable ways (CED 2.3.A): - If a > 0 and b > 1 (exponential growth), f(x) → ∞ (e.g., 2·3^x grows without bound). - If a > 0 and 0 < b < 1 (exponential decay), f(x) → 0⁺ (the graph approaches the horizontal asymptote y = 0 from above; e.g., 5·(1/2)^x → 0). - If a < 0 and b > 1, outputs go to −∞ as x → ∞ (negative growth). - If a < 0 and 0 < b < 1, outputs approach 0 from below. Remember: domain is all real numbers, range excludes 0 only when a ≠ 0 (outputs approach 0 but don’t equal it), graphs are monotonic and have no inflection points or interior extrema (CED 2.3.A.1–A.5). For more examples and AP-style practice, see the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and the unit practice page (https://library.fiveable.me/practice/ap-pre-calculus).
Can someone explain step by step how to identify if data shows exponential behavior?
Step-by-step: 1. Look at outputs over equal-length x-intervals (usually 1 unit). If the outputs change by a constant factor (multiplicative), it’s exponential. Example: 5, 10, 20 at x = 0,1,2 → each step ×2 so f(x)=5·2^x. 2. Compute successive ratios f(x+Δ)/f(x). If ratios are (approximately) constant for equal Δ, that suggests ab^x (CED 2.3.A.1 & 2.3.A.2). 3. Try a log transform: take ln(y). If ln(y) vs x is (approximately) linear, the original data are exponential. Fit y = ab^x with regression (on the AP exam Part B you can use a calculator for regressions). 4. Check shape: exponential functions are monotonic and always concave up or down (no inflection), and have horizontal asymptote y=0 (CED 2.3.A.3, 2.3.A.5). 5. If a constant vertical shift k is present, test g(x)=f(x)+k for proportionality over equal intervals (CED 2.3.A.4). For practice, see the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and more problems at (https://library.fiveable.me/practice/ap-pre-calculus).
Why is the domain of exponential functions all real numbers but the range isn't?
The domain is all real numbers because b^x is defined for any real x: b^x = e^{x ln b}, so you can plug in any real input and get a real output (with b>0, b≠1 per the CED). The range isn’t all reals because the outputs never cross zero—they’re always proportional to the initial value a. If a>0 then ab^x > 0 for every x, so the range is (0, ∞); if a<0 then ab^x < 0 for every x, so the range is (−∞, 0). That’s why exponential graphs have the horizontal asymptote y = 0 and no extrema (CED 2.3.A.1, 2.3.A.2, keywords: horizontal asymptote y=0, domain all real numbers, range positive/negative real numbers). For more practice and examples on this, check the Topic 2.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy) and the unit review (https://library.fiveable.me/ap-pre-calculus/unit-2).
What's the difference between exponential functions and power functions like x^2?
Short answer: they’re totally different places for the variable. An exponential has the variable in the exponent: f(x) = a b^x (initial value a, base b). A power (polynomial) like x^2 has the variable as the base with a fixed exponent. Key differences you should know for AP Topic 2.3:
- Behavior: ab^x is always increasing (b>1) or always decreasing (0
I missed class - how do I know if a word problem needs an exponential function?
Look for multiplicative/change-by-a-factor behavior. If a story says “grows/decays by X% each period,” “doubles,” “half-life,” or “changes by the same percent each interval,” it’s exponential—model it as f(x)=a·b^x (initial value a, base b>0,b≠1). The CED gives two quick tests: (1) outputs are proportional over equal-length input intervals (constant ratio), and (2) an additive shift g(x)=f(x)+k that still has proportional outputs implies f is exponential (2.3.A.3–2.3.A.4). Also note: exponentials are always increasing or always decreasing, have horizontal asymptote y=0, domain all reals, range positive (2.3.A.1, 2.3.A.2). Practical checklist: convert language to “multiply by … each unit”; check data for roughly constant successive ratios; if percent-per-period or compounding appears, use a(1+r)^t or A e^{kt} for continuous change. For AP modeling questions, items on Unit 2 often ask you to build or justify these models (see Topic 2.3 study guide: https://library.fiveable.me/ap-pre-calculus/unit-2/exponential-functions/study-guide/5hZXVBTYwi72vxCy). For extra practice, try problems at the unit page (https://library.fiveable.me/ap-pre-calculus/unit-2) or the practice bank (https://library.fiveable.me/practice/ap-pre-calculus).