APPC U2.5 Exponential Functions Context and Data Modeling

A bacteria culture starts with 500 bacteria and triples every 4 hours. Which function models the number of bacteria, $ B $, after $ t $ hours?

• B(t) = 500 \times 3^t

• B(t) = 500 \times 3^{t/4}

• B(t) = 500 \times (4^3)^t

• B(t) = 500 \times e^{3t}

Answer explanation

The bacteria triples every 4 hours, so after t hours, the number of 4-hour intervals is t/4. The model is B(t) = 500 * 3^(t/4), which correctly represents the growth pattern.

An investment of $1,200 is compounded continuously at an annual interest rate of 6%. Which function represents the amount, $ A $, after $ t $ years using the natural base $ e $?

• A(t) = 1200 \times (1.06)^t

• A(t) = 1200 \times e^{0.06t}

• A(t) = 1200 \times e^{6t}

• A(t) = 1200 + 0.06t

Answer explanation

The formula for continuous compounding is A(t) = Pe^{rt}, where P is the principal, r is the rate, and t is time. Here, P = 1200 and r = 0.06, leading to A(t) = 1200 × e^{0.06t}, which is the correct choice.

The population of a town grows according to the function $ P(t) = 3000 \times 1.04^t $, where $ t $ is in years. What is the annual growth rate in percentage?

• 1%

• 4%

• 40%

• 0.4%

Answer explanation

The population function is given by P(t) = 3000 × 1.04^t. The base of the exponent, 1.04, indicates an annual growth rate of 4%. Thus, the correct answer is 4%.

Given two points $(1, 150)$ and $(4, 1200)$ on an exponential function $ f(x) = ab^x $, find the value of $ b $.

• 2

• 2^{1/3}

• 3

• \sqrt{2}

Answer explanation

To find $ b $, use the points in the equation $ f(x) = ab^x $. From $ f(1) = 150 $ and $ f(4) = 1200 $, we get two equations: $ ab = 150 $ and $ ab^4 = 1200 $. Dividing gives $ b^3 = 8 $, so $ b = 2 $.

A function $ f(x) = 250 \times e^{0.04x} $ represents continuous growth. What is the growth rate expressed as a percentage?

• 0.04%

• 4%

• 40%

• 400%

Answer explanation

The function f(x) = 250 × e^(0.04x) indicates continuous growth with a growth rate of 0.04. To express this as a percentage, multiply by 100, resulting in 4%. Thus, the correct answer is 4%.

To model a scenario where the amount of money $ M(t) $ is equal to 800 multiplied by $ 1.05^t $ plus 150, which transformation has been applied to the basic exponential function?

• Vertical shift upward by 150

• Horizontal shift by 150

• Vertical stretch by 150

• Reflection over the x-axis

Answer explanation

The function M(t) = 800 * 1.05^t + 150 shows a vertical shift upward by 150 units. The term +150 indicates that the entire graph of the basic exponential function is moved up by 150, making this the correct transformation.

Which of the following functions could represent a proportional growth situation where the output is multiplied by 4 every 5 time units, starting with an initial value of 100?

• f(t) = 100 \times 4^t

• f(t) = 100 \times 4^{t/5}

• f(t) = 100 + 4t

• f(t) = 100 \times t^4

Answer explanation

The function f(t) = 100 × 4^{t/5} correctly represents the situation, as it indicates that the output is multiplied by 4 every 5 time units, starting from 100. The other options do not reflect this proportional growth.

A function $ f(x) = ab^x $ has an initial value of 60 and passes through the point $(2, 180)$. What is the value of $ a $?

• 60

• 180

• 3

• It cannot be determined from the given information.

Answer explanation

The function has an initial value of 60, meaning when x=0, f(0)=a=60. Thus, the value of a is 60.

If $ f(x) = 150 \times 2^{x} $ and $ g(x) = f(x) + 50 $, which transformation relates $ g $ to $ f $?

• g(x) is a horizontal shift of f(x) by 50 units.

• g(x) is a vertical shift of f(x) by 50 units upward.

• g(x) is a vertical stretch of f(x) by a factor of 50.

• g(x) reflects f(x) over the x-axis and shifts it up by 50 units.

Answer explanation

g(x) = f(x) + 50 indicates that g(x) is obtained by adding 50 to the output of f(x). This means g(x) is a vertical shift of f(x) by 50 units upward, making the correct choice the second option.

An exponential function models the depreciation of a car's value. If the value decreases by 20% each year and the initial value is $25,000$, which function represents the car’s value, $ V $, after $ t $ years?

• V(t) = 25000 \times 0.8^t

• V(t) = 25000 \times 1.2^t

• V(t) = 25000 - 0.2t

• V(t) = 25000 \times e^{-0.2t}

Answer explanation

The car's value decreases by 20% each year, meaning it retains 80% of its value. The function V(t) = 25000 × 0.8^t correctly represents this exponential decay, where 0.8 is the remaining value after each year.