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Questions and Answers
The population growth is modeled using an ______ function, with the starting value of 5.4 million people.
The population growth is modeled using an ______ function, with the starting value of 5.4 million people.
exponential
The equation is written as y = 5.4 × 1.01^x, where 1.01 is the decimal form of ______ percent.
The equation is written as y = 5.4 × 1.01^x, where 1.01 is the decimal form of ______ percent.
101
The depreciation rate represents an ______ decay, with the value decreasing by 12% each year.
The depreciation rate represents an ______ decay, with the value decreasing by 12% each year.
exponential
The constant multiple for the exponential decay equation is 0.88, calculated by subtracting the depreciation rate from ______ percent.
The constant multiple for the exponential decay equation is 0.88, calculated by subtracting the depreciation rate from ______ percent.
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The equation for the car's value over time is y = 36,000 × 0.88^x, where x represents the number of ______.
The equation for the car's value over time is y = 36,000 × 0.88^x, where x represents the number of ______.
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What is the purpose of the exponential function in the population growth model?
What is the purpose of the exponential function in the population growth model?
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What is the value of the car after six years, rounded to the nearest dollar?
What is the value of the car after six years, rounded to the nearest dollar?
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What is the starting value of the population in the exponential growth model?
What is the starting value of the population in the exponential growth model?
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What is the decimal form of the annual depreciation rate?
What is the decimal form of the annual depreciation rate?
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What is the population after 20 years, rounded to the nearest one-tenth of a million?
What is the population after 20 years, rounded to the nearest one-tenth of a million?
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Study Notes
- The function f(x) = -2 × 3^x is evaluated, and a pattern is spotted by plugging in values into a calculator.
- The y-intercept is found to be -2, which is the value of f(0).
- The pattern of multiplying by 3 is observed, and the next values are calculated: f(1) = -6, f(2) = -18, and so on.
- The pattern also holds for negative exponents, with f(-1) = -2/3 or -0.67, and f(-2) = -2/9 or -0.22.
- The graph of the function is plotted, showing the exponential growth and an asymptote on the left side.
- A table of values is given, but no equation, and the task is to find the equation.
- The pattern in the table is multiplying by 6, indicating an exponential function, which can be written in the form y = a × b^x.
- The y-intercept (a) is found by using the pattern to go back a couple of steps, resulting in a = 26/3 or 8.67.
- The equation is written as y = 13/9 × 6^x.
- Another equation is given, with a y-intercept in the middle, and the pattern is to divide by 7, resulting in the equation y = 50 × (1/7)^x.
- The equation is rewritten to fit the format y = a × b^x, with a = 50 and b = 1/7.
- A story problem is given, where a person has $2680 and earns 1.6% interest annually, compounded annually.
- The equation is written as y = 2680 × 1.016^x, where 1.016 is the decimal form of 101.6%.
- The value of the account after 10 years is calculated to be $3141.03.
- A question about Colorado's population is given, which grew 1% from July 2015 to July 2016.
- The population growth is modeled using an exponential function, with the starting value of 5.4 million people.
- The equation is written as y = 5.4 × 1.01^x, where 1.01 is the decimal form of 101%.
- The population after 20 years is calculated using the equation.- Total population calculated to be 6.6 million people, rounded to the nearest one-tenth of a million.
- A new car was purchased in 2016 for $36,000, with an annual depreciation rate of 12%.
- The depreciation rate represents an exponential decay, with the value decreasing by 12% each year.
- The constant multiple for the exponential decay equation is 0.88, calculated by subtracting the depreciation rate from 100%.
- The equation for the car's value over time is y = 36,000 × 0.88^x, where x represents the number of years.
- After six years, the value of the car is calculated to be $16,719, rounded to the nearest dollar.
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Description
Learn how to identify and work with exponential functions, including exponential growth and decay, and apply them to real-world scenarios such as population growth, compound interest, and depreciation.
