Exponential Functions: Practice Questions & Study Guide
Modeling growth and decay with exponential functions, interpreting the base and exponent, and comparing exponential to linear growth.
Understanding Exponential Functions
An exponential function has the form f(x) = a * b^x, where a is the initial value (f(0) = a) and b is the growth factor per unit increase in x. When b > 1, the function models growth (e.g., population increase, compound interest); when 0 < b < 1, it models decay (e.g., radioactive decay, depreciation). The test tests your ability to interpret these parameters in context, write exponential models from given information, and compare exponential to linear growth.
The growth rate r relates to the base by b = 1 + r for growth (e.g., 5% annual growth gives b = 1.05) or b = 1 - r for decay (e.g., 20% annual depreciation gives b = 0.80). A question might give you the formula P(t) = 2000 * 1.06^t and ask what the 1.06 represents—the answer is that the population increases by 6% per year (the growth factor). Being fluent in this translation between the algebraic base and the percentage change is essential for exponential questions.
The test also tests the starting value (a) and what it represents in context. For f(t) = 500 * 0.85^t modeling the value of a car after t years, a = 500 is the initial value ($500 at t = 0) and b = 0.85 means the car retains 85% of its value each year (depreciates by 15% per year). Questions may present this with different context (bacteria, investment, medication concentration) but the interpretation is always the same.
Comparing exponential and linear growth is another common theme: for small x values, a linear function with a steep slope may exceed an exponential; but for large x values, exponential growth always outpaces any linear function. Questions might give you a table of values and ask which model (linear or exponential) fits, or ask beyond which point the exponential model produces larger values. For linear fit, look for constant differences; for exponential fit, look for constant ratios between consecutive values.
Key Rules & Formulas
Memorize these rules — they come up directly in practice questions.
In f(x) = a * b^x, a is the initial value (f(0) = a) and b is the growth/decay factor per period.
f(t) = 400 * 1.08^t: initial value is 400; each period the quantity multiplies by 1.08 (grows 8%).
If b = 1 + r, then r is the growth rate per period (as a decimal). For decay, b = 1 - r.
b = 0.92 means 8% decay per period (r = 0.08, b = 1 - 0.08).
Exponential data has a constant ratio (multiplier) between consecutive terms; linear data has a constant difference.
Values 5, 10, 20, 40 have constant ratio 2 → exponential model. Values 5, 10, 15, 20 have constant difference 5 → linear model.
To find the equation from two data points, solve for a and b using the ratio of the two outputs.
If f(0) = 100 and f(3) = 800: 100 * b^3 = 800, b^3 = 8, b = 2. So f(x) = 100 * 2^x.
f(x + 1) / f(x) = b for any exponential function, confirming constant ratio.
For f(x) = 3 * 5^x: f(2)/f(1) = 75/15 = 5 = b. ✓
Exponential Functions Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
A bacteria population starts at 500 and doubles every hour. Which function models the population P after t hours?
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Correct answer: C. P(t) = 500 * 2^t
Explanation
Doubling each hour means multiplying by 2 each hour. Starting at 500, after t hours: P(t) = 500 * 2^t.
In the function f(t) = 1200 * (0.75)^t, what does 0.75 represent?
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Correct answer: C. The function retains 75% of its value each unit of t
Explanation
The base 0.75 is the decay factor. Each unit increase in t multiplies the value by 0.75, meaning the quantity retains 75% of its value (and loses 25%) per period.
If f(x) = 4 * 3^x, what is f(2)?
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Correct answer: B. 36
Explanation
f(2) = 4 * 3^2 = 4 * 9 = 36.
The value of an investment after t years is V(t) = 2500 * (1.06)^t dollars. What does 1.06 represent in this context?
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Correct answer: B. The investment earns 6% interest per year
Explanation
The base 1.06 = 1 + 0.06 = 1 + 6%. This means the investment grows by 6% per year (the annual interest rate is 6%).
A radioactive substance decays according to A(t) = 800 * (0.5)^(t/3), where t is in years. What is the half-life of the substance?
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Correct answer: C. 3 years
Explanation
The half-life is the time for the substance to decrease to half its current amount. When t = 3: A(3) = 800 * (0.5)^1 = 400, which is half of 800. So the half-life is 3 years.
Which of the following tables shows values consistent with an exponential (not linear) function?
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Correct answer: B. x: 0,1,2,3 and f(x): 3,6,12,24
Explanation
Exponential functions have a constant ratio between consecutive values. Choice B: 6/3 = 2, 12/6 = 2, 24/12 = 2 ✓. Choices A, C, and D have constant differences (linear functions).
An exponential function f satisfies f(0) = 100 and f(2) = 400. Which equation represents f?
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Correct answer: A. f(x) = 100 * 2^x
Explanation
f(0) = 100 ✓ for A, B, C. For f(2) = 400: A: 100*4 = 400 ✓; B: 100*16 = 1600 ✗; C: 100*2 = 200 ✗. So f(x) = 100 * 2^x.
A town's population grows according to P(t) = P_0 * b^t. In 2010 the population was 8,000; in 2014 it was 12,500. What is the value of b (the annual growth factor)?
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Correct answer: A. b = (12500/8000)^(1/4)
Explanation
From 2010 to 2014 is 4 years: P_0 * b^4 = 12500 with P_0 = 8000. So b^4 = 12500/8000 = 25/16 → b = (25/16)^(1/4) = (12500/8000)^(1/4).
If f(x) = a * b^x and f(1) = 6 and f(3) = 54, what is f(5)?
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Correct answer: C. 486
Explanation
From f(3)/f(1) = (a*b^3)/(a*b^1) = b^2 = 54/6 = 9, so b = 3. From f(1) = 6: a*3 = 6, so a = 2. Thus f(x) = 2 * 3^x. f(5) = 2 * 3^5 = 2 * 243 = 486.
The functions f(x) = 2x + 100 and g(x) = 2^x both start with g(0) < f(0). For what integer value of x does g(x) first exceed f(x)?
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Correct answer: C. x = 7
Explanation
Compare values: x=6: f=112, g=64. x=7: f=114, g=128. At x=7, g(7) = 128 > f(7) = 114 for the first time. So x = 7 is the first integer where g exceeds f.
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Common Mistakes to Avoid
These are the most frequent errors students make on Exponential Functions questions. Knowing them in advance prevents costly point losses.
- !Confusing the initial value (a, when x = 0) with the growth factor (b, the base).
- !Writing b = 1.15 when the problem says the quantity 'decreases by 15%'—it should be b = 0.85.
- !Evaluating exponential expressions with order-of-operations errors: 3 * 2^4 is 3 * 16 = 48, NOT 6^4 = 1296.
- !Thinking exponential decay eventually reaches zero—it approaches zero asymptotically but never actually reaches it.
- !Confusing 'doubles every 3 years' (which means f(t) = a * 2^(t/3)) with 'doubles every year' (f(t) = a * 2^t).
Strategy Tips: Exponential Functions
When a question says a quantity 'increases by p% per year,' the base is (1 + p/100) and the model is f(t) = a(1 + p/100)^t.
Check whether data in a table is exponential by computing successive ratios—if constant, write the exponential model directly from the ratio (b) and first term (a).
For questions asking what a specific number in the formula represents, translate: 'the base 1.03 represents a 3% growth per period.'
Use the graphing calculator to evaluate specific outputs of exponential functions—do not attempt exponential arithmetic by hand when precision matters.
Other Advanced Math Subtopics
Equivalent Expressions
Rewriting algebraic expressions into equivalent forms through factoring, expanding, and applying algebraic identities.
Nonlinear Equations in One Variable
Solving quadratic, radical, and rational equations, and understanding the conditions under which extraneous solutions arise.
Nonlinear Functions
Understanding polynomial, radical, and rational functions—evaluating them, identifying their key features, and interpreting them in context.
Quadratic Equations and Parabolas
Solving quadratics, identifying the vertex and intercepts of parabolas, and interpreting these features in applied contexts.
Master Exponential Functions
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