10 Questions
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.
101
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.
100
The equation for the car's value over time is y = 36,000 × 0.88^x, where x represents the number of ______.
years
What is the purpose of the exponential function in the population growth model?
To model the growth of the population over time
What is the value of the car after six years, rounded to the nearest dollar?
$16,719
What is the starting value of the population in the exponential growth model?
5.4 million people
What is the decimal form of the annual depreciation rate?
0.12
What is the population after 20 years, rounded to the nearest one-tenth of a million?
6.6 million people
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.
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.
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