Exponential Functions and Growth
Modeling patterns of repeated multiplication
Exponential Functions and Growth
Modeling patterns of repeated multiplication
Math - Grade 9-12
- 1
Evaluate the function y = 3(2^x) when x = 4.
Substitute 4 for x and evaluate the exponent first.
When x = 4, y = 3(2^4) = 3(16) = 48. The value of the function is 48. - 2
Evaluate f(x) = 5(1.2^x) for x = 3. Round your answer to the nearest tenth.
When x = 3, f(3) = 5(1.2^3) = 5(1.728) = 8.64. Rounded to the nearest tenth, the value is 8.6. - 3
Determine whether the function y = 7(0.6^x) represents exponential growth or exponential decay. Explain how you know.
Look at whether the base is greater than 1 or between 0 and 1.
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. - 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.
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. - 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.
A percent increase uses 1 + rate as the growth factor.
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. - 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.
After 3 years, V(3) = 24000(0.85^3) = 24000(0.614125) = 14739. The value of the car is about $14,739. - 7
For the exponential function y = 2(3^x), find the y-intercept.
Set x equal to 0 to find the y-intercept.
The y-intercept occurs when x = 0. Then y = 2(3^0) = 2(1) = 2. The y-intercept is (0, 2). - 8
Compare the functions y = 4(2^x) and y = 4(5^x). Which function grows faster as x increases, and why?
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. - 9
Rewrite the percent growth statement as an exponential factor: a quantity increases by 12% each year.
Convert the percent to a decimal and add it to 1.
The exponential growth factor is 1.12. A 12% increase means adding 0.12 to 1. - 10
Rewrite the percent decrease statement as an exponential factor: a quantity decreases by 18% each month.
The exponential decay factor is 0.82. An 18% decrease means subtracting 0.18 from 1. - 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.
Use initial amount times growth factor to the power of time.
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. - 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?
The initial value is 900 because f(0) = 900(1.5^0) = 900. This means the video starts with 900 views at time 0. - 13
Which function represents exponential growth: y = 6x + 2 or y = 6(1.2^x)? Explain your choice.
Exponential functions have the variable in the exponent.
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. - 14
A medicine dose is reduced by 25% each hour. If the starting amount is 80 milligrams, how much remains after 2 hours?
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. - 15
Write an exponential function for the table where x = 0, 1, 2, 3 and y = 6, 12, 24, 48.
Look for the starting value and the repeated multiplication pattern.
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.