Exponential Functions and Exponent Rules · Sub-skill drill
Comparing Exponential and Linear Models
An exponential function eventually grows faster than any linear function, but linear can be larger in the short term. SAT questions sometimes ask which model fits a given table or scenario better. Linear models have constant differences between consecutive y-values; exponential models have constant ratios between consecutive y-values. Computing the first differences and the first ratios across the table is the fastest classification method.
How this sub-skill is tested on the SAT
An exponential function eventually grows faster than any linear function, but linear can be larger in the short term. SAT questions sometimes ask which model fits a given table or scenario better. Linear models have constant differences between consecutive y-values; exponential models have constant ratios between consecutive y-values. Computing the first differences and the first ratios across the table is the fastest classification method.
This sub-skill sits inside the broader Exponential Functions and Exponent Rules topic, which is part of the College Board's Passport to Advanced Math content domain. Passport to Advanced Math is the SAT's bridge to the kind of algebraic manipulation you will see in a college precalculus or calculus course. The questions are not about memorizing identities — they are about fluency with structure. Can you factor a quadratic by inspection? Can you read the vertex of a parabola off its standard form? Can you simplify a rational expression without losing a domain restriction? Can you translate between f(x), a graph, and a table of values without panicking? Most students who plateau between 650 and 720 plateau here, because the section rewards algebraic intuit
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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A radioactive sample of mass 514 grams loses a factor of 10 every year. What is its mass after 4 years?
- A 474.00
- B 0.51
- C 0.05
- D 12.85
Worked solution
Answer: C — 0.05
Exponential decay: m(t) = m_0 × (1/10)^t = 514 / 10^4 = 514 / 10000 = 0.05 grams.
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A radioactive sample of mass 712 grams loses a factor of 10 every year. What is its mass after 3 years?
- A 7.12
- B 682.00
- C 0.71
- D 23.73
Worked solution
Answer: C — 0.71
Exponential decay: m(t) = m_0 × (1/10)^t = 712 / 10^3 = 712 / 1000 = 0.71 grams.
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A radioactive sample of mass 755 grams loses a factor of 10 every year. What is its mass after 1 years?
- A 755.00
- B 75.50
- C 745.00
- D 76.5
Worked solution
Answer: B — 75.50
Exponential decay: m(t) = m_0 × (1/10)^t = 755 / 10^1 = 755 / 10 = 75.50 grams.
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A radioactive sample of mass 331 grams loses a factor of 10 every year. What is its mass after 1 years?
- A 321.00
- B 331.00
- C 33.10
- D 34.1
Worked solution
Answer: C — 33.10
Exponential decay: m(t) = m_0 × (1/10)^t = 331 / 10^1 = 331 / 10 = 33.10 grams.
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A radioactive sample of mass 575 grams loses a factor of 10 every year. What is its mass after 1 years?
- A 58.5
- B 575.00
- C 565.00
- D 57.50
Worked solution
Answer: D — 57.50
Exponential decay: m(t) = m_0 × (1/10)^t = 575 / 10^1 = 575 / 10 = 57.50 grams.
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A radioactive sample of mass 699 grams loses a factor of 10 every year. What is its mass after 2 years?
- A 34.95
- B 69.90
- C 679.00
- D 6.99
Worked solution
Answer: D — 6.99
Exponential decay: m(t) = m_0 × (1/10)^t = 699 / 10^2 = 699 / 100 = 6.99 grams.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.