Exponential Functions and Growth Worksheet

Question 1

Evaluate the function y = 3(2^x) when x = 4.

Accepted Answer

When x = 4, y = 3(2^4) = 3(16) = 48. The value of the function is 48.

Question 2

Evaluate f(x) = 5(1.2^x) for x = 3. Round your answer to the nearest tenth.

Accepted Answer

When x = 3, f(3) = 5(1.2^3) = 5(1.728) = 8.64. Rounded to the nearest tenth, the value is 8.6.

Question 3

Determine whether the function y = 7(0.6^x) represents exponential growth or exponential decay. Explain how you know.

Accepted Answer

The function represents exponential decay because the base 0.6 is between 0 and 1. In an exponential function, a base between 0 and 1 causes the values to decrease as x increases.

Question 4

A bacteria culture starts with 200 cells and doubles every hour. Write an exponential function that models the number of cells after t hours.

Accepted Answer

An exponential model is N(t) = 200(2^t). The initial amount is 200 and the growth factor is 2 because the culture doubles each hour.

Question 5

A savings account starts with $500 and grows by 4% each year. Write an exponential function that gives the account balance after t years.

Accepted Answer

An exponential growth model is A(t) = 500(1.04^t). The initial value is 500 and the growth factor is 1 + 0.04 = 1.04.

Question 6

The value of a car is modeled by V(t) = 24000(0.85^t), where t is the number of years after purchase. Find the value of the car after 3 years. Round to the nearest dollar.

Accepted Answer

After 3 years, V(3) = 24000(0.85^3) = 24000(0.614125) = 14739. The value of the car is about $14,739.

Question 7

For the exponential function y = 2(3^x), find the y-intercept.

Accepted Answer

The y-intercept occurs when x = 0. Then y = 2(3^0) = 2(1) = 2. The y-intercept is (0, 2).

Question 8

Compare the functions y = 4(2^x) and y = 4(5^x). Which function grows faster as x increases, and why?

Accepted Answer

The function y = 4(5^x) grows faster because it has the larger base. For exponential growth functions with the same initial value, the function with the larger base increases more quickly.

Question 9

Rewrite the percent growth statement as an exponential factor: a quantity increases by 12% each year.

Accepted Answer

The exponential growth factor is 1.12. A 12% increase means adding 0.12 to 1.

Question 10

Rewrite the percent decrease statement as an exponential factor: a quantity decreases by 18% each month.

Accepted Answer

The exponential decay factor is 0.82. An 18% decrease means subtracting 0.18 from 1.

Question 11

A population of 1,500 grows by 3% per year. About how many people are in the population after 5 years? Round to the nearest whole number.

Accepted Answer

The model is P(t) = 1500(1.03^t). After 5 years, P(5) = 1500(1.03^5) about 1500(1.159274) = 1738.911. Rounded to the nearest whole number, the population is about 1,739 people.

Question 12

The function f(x) = 900(1.5^x) models the number of views of a video over time. What is the initial value, and what does it mean in this context?

Accepted Answer

The initial value is 900 because f(0) = 900(1.5^0) = 900. This means the video starts with 900 views at time 0.

Question 13

Which function represents exponential growth: y = 6x + 2 or y = 6(1.2^x)? Explain your choice.

Accepted Answer

The function y = 6(1.2^x) represents exponential growth because the variable is in the exponent and the base 1.2 is greater than 1. The function y = 6x + 2 is linear, not exponential.

Question 14

A medicine dose is reduced by 25% each hour. If the starting amount is 80 milligrams, how much remains after 2 hours?

Accepted Answer

A 25% decrease means the decay factor is 0.75, so the model is A(t) = 80(0.75^t). After 2 hours, A(2) = 80(0.75^2) = 80(0.5625) = 45. The amount remaining is 45 milligrams.

Question 15

Write an exponential function for the table where x = 0, 1, 2, 3 and y = 6, 12, 24, 48.

Accepted Answer

An exponential function for the table is y = 6(2^x). The initial value is 6 and the output doubles each time x increases by 1, so the growth factor is 2.

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