SAT Math Quiz: Linear And Exponential Growth

What this quiz covers

This quiz focuses on Linear And Exponential Growth, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $N(t)=500(2)^{t}$
2. $N(t)=500(2)^{t/3}$ (correct answer)
3. $N(t)=500+2t$
4. $N(t)=500(3)^{t/2}$

Explanation: We need to find an equation for bacteria that start at 500 cells and double every 3 hours. Since the population multiplies by a constant factor (doubles), this is exponential growth. The general form is N(t) = N₀ · (growth factor)^(t/doubling time), where N₀ = 500, growth factor = 2, and doubling time = 3 hours. This gives us N(t) = 500(2)^(t/3). We can verify: at t = 0, N(0) = 500(2)^0 = 500; at t = 3, N(3) = 500(2)^1 = 1000 (doubled). A common mistake is using N(t) = 500(2)^t, which would mean doubling every hour instead of every 3 hours.

Question 2

1. -\frac{1}{3}
2. 3
3. \frac{1}{3}
4. -3 (correct answer)

Explanation: Since $\frac{1}{8}=2^{-3}$, it follows that $x=-3$. The other choices come from ignoring the reciprocal or misinterpreting exponents.

Question 3

1. Adds 1.5 cells/hr; linear growth
2. Multiplies by 1.5 each hr; exponential growth (correct answer)
3. Adds 50% cells/hr; linear growth
4. Multiplies by 50 each hr; exponential growth

Explanation: The question asks for the interpretation of the parameter 1.5 in the exponential equation N(t) = 200(1.5)^t. This equation follows the form of exponential growth where the base (1.5) represents the growth factor. In exponential models, the population is multiplied by the growth factor each time period, so the bacteria population multiplies by 1.5 each hour. This means the population increases by 50% each hour (since 1.5 = 1 + 0.50). The key error would be confusing multiplicative growth with additive growth - exponential functions multiply by a constant factor, not add a constant amount.

Question 4

1. $3\%$ (correct answer)
2. $30\%$
3. $103\%$
4. $0.03\%$

Explanation: We need to find the growth rate from the model W(t) = 8000(1.03)^t. In exponential growth models of the form y = a(1+r)^t, the growth rate is r. Here, 1.03 = 1 + 0.03, so r = 0.03 = 3%. This means water usage grows by 3% each year. To verify: if usage grows by 3%, then after one year it becomes 8000 × 1.03, which matches our model at t=1. Don't confuse the growth factor (1.03) with the growth rate (0.03 or 3%). The growth factor is always 1 plus the growth rate for exponential growth.

Question 5

1. 2 (correct answer)
2. 4
3. -2
4. 8

Explanation: From $(0,6)$ we get $a=6$, and $6b^2=24$ gives $b^2=4$ so $b=2$ (base positive). Choosing 4 confuses the two-step ratio with the one-step base, and $-2$ is invalid for an exponential base.

Question 6

1. $y = 60x$ (correct answer)
2. $y = 60^x$
3. $y = 60 + x$
4. $y = x/60$

Explanation: This problem asks which equation represents distance traveled at a constant speed of 60 mph. Since distance equals rate times time when speed is constant, this describes a linear relationship. The equation is y = 60x, where y represents total distance and x represents hours traveled.

Question 7

1. 100
2. 75
3. 25
4. 50 (correct answer)

Explanation: Here $t/3=6/3=2$, so $N(6)=200(0.5)^2=200\cdot0.25=50$. The other choices result from taking one half-life instead of two or misreading the exponent.

Question 8

1. $2.75$ base fare; $21.00$
2. $4.50$ per mile; $21.00$
3. $2.75$ per mile; $21.00$ (correct answer)
4. $4.50$ base fare; $18.00$

Explanation: The question provides the taxi fare equation y = 2.75x + 4.50 and asks for the meaning of the slope and the cost of a 6-mile ride. In this linear equation, the slope 2.75 represents the rate of change - the cost per mile ($2.75/mile). The y-intercept 4.50 is the base fare. For a 6-mile ride: y = 2.75(6) + 4.50 = 16.50 + 4.50 = 21.00 dollars. The key concept is interpreting slope as rate of change in context - here it's the variable cost per unit distance. Common errors include confusing the roles of slope and y-intercept in real-world applications.

Question 9

1. $B(t)=100(0.88)^t$
2. $B(t)=12t-100$
3. $B(t)=100+12t$
4. $B(t)=100-12t$ (correct answer)

Explanation: A battery starts at 100% and loses 12 percentage points per hour. "Loses 12 percentage points" indicates subtraction (linear decay), not multiplication (exponential decay). The model is B(t) = 100 - 12t, where we subtract 12t from the initial 100%. We can verify: at t=1, B(1) = 100-12(1) = 88%; at t=2, B(2) = 100-12(2) = 76%. The key distinction is between losing a constant amount (linear) versus losing a constant percentage of the remaining amount (exponential).

Question 10

1. Linear; straight-line trend
2. Exponential; increasing rate of change (correct answer)
3. Linear; constant ratio each year
4. Exponential; constant difference each year

Explanation: This question describes a graph that rises slowly at first and then more rapidly, which is characteristic of exponential growth. Exponential functions show an increasing rate of change - the curve gets steeper as time progresses because each increase is a percentage of the current value. Linear functions maintain a constant rate of change and would appear as a straight line. The key feature supporting exponential growth is the accelerating rate of increase. A common error is choosing linear because of confusion about what "constant ratio" means. For visual identification: if the curve bends upward (concave up), it's likely exponential growth.

Question 11

1. Linear; differences are constant
2. Exponential; ratios are constant (correct answer)
3. Linear; ratios are constant
4. Exponential; differences are constant

Explanation: This problem gives three points to analyze: (0, 40), (1, 60), and (2, 90). Checking differences: 60 - 40 = 20 and 90 - 60 = 30, which are not constant. Checking ratios: 60/40 = 1.5 and 90/60 = 1.5, which are constant. Since the ratios are constant at 1.5, this represents exponential growth where revenue increases by 50% each hour. The model would be R = 40(1.5)^t. The key insight is that constant ratios indicate exponential growth, while constant differences indicate linear growth. Always check both to determine the appropriate model type.

Question 12

1. $750$/week; $14,000$ (correct answer)
2. $750$ initial; $14,000$
3. 15%/week; $11,000$
4. $5000$/week; $14,000$

Explanation: This problem gives us a linear model $D(t) = 5000 + 750t$ for charity donations. In this model, 5000 represents the initial amount (when t = 0), and 750 represents the rate of increase per week. After 12 weeks: $D(12) = 5000 + 750(12) = 5000 + 9000 = 14,000$ dollars. The parameter 750 means donations increase by $750$ each week (constant additive growth). A common error is misidentifying which parameter represents what - remember that in $y = mx + b$, m is the rate of change and b is the initial value. For time-based models, the coefficient of t is always the rate per time unit.

Question 13

1. $2.2$
2. $1.6$
3. $3.6$
4. $1.728$ (correct answer)

Explanation: The factor over 3 units is $(1.2)^3=1.728$. Choices B, C, and D incorrectly add percentages or mix factor with rate.

Question 14

1. Population decreases by 85 each year
2. Population keeps 85% each year (correct answer)
3. Initial population is 0.85
4. Population decreases by 0.85 each year

Explanation: This question asks us to interpret the base 0.85 in the exponential model P(t) = 1200(0.85)^t. In exponential models of the form y = a(b)^t, the base b represents the growth/decay factor - what the quantity is multiplied by each time period. Since 0.85 < 1, this represents decay, and 0.85 means the population retains 85% of its value each year (or loses 15%). The population doesn't decrease BY 0.85 or BY 85%; rather, it's multiplied by 0.85. Understanding the difference between "decreases by" (subtraction) and "decreases to" (multiplication) is crucial for interpreting exponential models.

Question 15

1. $N=200(2)^t$
2. $N=200(2)^{t/3}$ (correct answer)
3. $N=200+2(t/3)$
4. $N=200(3)^{t/2}$

Explanation: This problem involves modeling exponential growth with a non-unit time factor. Since the culture doubles every 3 hours, after t hours it has gone through t/3 doubling periods. Starting with 200 cells, the model is N = 200(2)^(t/3), which correctly accounts for the 3-hour doubling time. For example, after 6 hours (2 doubling periods), N = 200(2)^2 = 800 cells. Students often write N = 200(2)^t, forgetting to adjust for the doubling period. When the growth period doesn't match the time unit, always include the appropriate fraction in the exponent. Test your model at easy values like t = 3 to verify correctness.

Question 16

1. Quadratic
2. Linear
3. Exponential (correct answer)
4. Logarithmic

Explanation: This question asks us to classify the type of growth shown by a sequence of numbers. To determine growth type, we check if consecutive terms have constant differences (linear) or constant ratios (exponential). Looking at the sequence 5, 10, 20, 40, let's find the ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2. Since each term is exactly double the previous term, this demonstrates exponential growth with a base of 2. The multiplicative pattern (×2 each time) rather than an additive pattern (+5 each time) confirms exponential behavior. A common error would be to see the doubling and think "it's just adding itself," but exponential growth is characterized by constant multiplicative factors. When you see consistent multiplication or division between terms, think exponential.

Question 17

1. The candle burns $12$ inches per hour
2. The candle height increases to $12$ inches
3. The initial height is $12$ inches (correct answer)
4. The candle lasts $12$ hours

Explanation: The candle height model h(t) = 12 - 0.8t is linear, where t is hours since lighting. In the form y = mx + b, the constant term (12) represents the y-intercept, which is the value when t = 0. Therefore, 12 represents the initial height of the candle in inches before any burning occurs. We can verify: at t = 0, h(0) = 12 - 0.8(0) = 12 inches. The coefficient -0.8 tells us the candle burns 0.8 inches per hour. A common error is confusing the constant term with the rate of change or the time duration. In linear models, always identify what happens at time zero to understand the constant term.

Question 18

1. 0.80%
2. 108%
3. 0.08%
4. 8% (correct answer)

Explanation: The growth factor is 1.08, which corresponds to an 8% increase. 108% confuses the factor with the rate, and 0.08% or 0.8% are decimal-place errors.

Question 19

1. 200
2. 800 (correct answer)
3. 400
4. 900

Explanation: This problem asks for the population after 9 hours when bacteria doubles every 3 hours. Since the population doubles at regular intervals, this follows an exponential growth model. Starting with 100 bacteria, after 3 hours we have 100 × 2 = 200, after 6 hours we have 200 × 2 = 400, and after 9 hours we have 400 × 2 = 800.

Question 20

1. $-\frac{3}{2}$
2. $1$
3. $3$
4. $\frac{3}{2}$ (correct answer)

Explanation: $3^{2x}=27=3^3$, so $2x=3$ and $x=\frac{3}{2}$. A negative root is extraneous because $3^x>0$ for all real $x$.