Questions and Answers
Which graph shows exponential growth?
Which is the graph of f(x) = 1/4 (4)x?
What equation can be used to predict, y, the number of people living in the town after x years? (Population values are rounded to the nearest whole number.)
y = 32,000(1.08)x
What are the domain and range of the function on the given graph?
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Which equation represents y, the value of the item after x years?
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Which is the graph of f(x) = 0.5(4)x?
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Compare and contrast the domain and range for the functions f(x) = 5x and g(x) = 5x.
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Which equation represents an exponential function that passes through the point (2, 36)?
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Which equation represents an exponential function that passes through the point (2, 80)?
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Which statements are true about exponential functions? Check all that apply.
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Which equation represents the amount of money in Josiah's account, y, after x years?
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What are the domain and range of the function based on the ordered pairs from the continuous exponential function?
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What is the multiplicative rate of change of the function f(x) = 2(5)x?
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Which graph represents the initial step of plotting the function f(x) = 3(4)x?
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What are the domain and range of the function shown on the given graph?
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Which is the graph of f(x) = 5(2)x?
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What is the rate of change of the function described in the table?
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What is the rate of change of the function shown on the graph? Express your answer in decimal form rounded to the nearest tenth.
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Study Notes
Exponential Growth Functions Overview
- Exponential growth is depicted on a specific type of graph that shows a rapid increase over time.
- An example function demonstrating exponential growth is f(x) = 1/4 (4)x.
Population Growth Example
- A town's population increases exponentially from 34,560 after 1 year to 37,325 after 2 years.
- The predictive formula for future population growth is y = 32,000(1.08)x.
Value Increase of Collectibles
- A collector's item valued at $500 appreciates at a rate of 5% annually.
- The equation representing its value after x years is y = 500(1.05)x, reaching $551.25 after 2 years.
Graphs of Exponential Functions
- The graph of f(x) = 0.5(4)x shows exponential behavior.
- It’s important to understand how various functions can look visually.
Domain and Range of Exponential Functions
- For exponential functions like f(x) = 5x and g(x) = 5x, the domain is all real numbers with ranges differing (y > 0 for g(x)).
- A common characteristic of exponential functions is the domain set to all real numbers and the range being positive (y > 0).
Defining Exponential Functions
- An exponential function can pass through specific points; for example, f(x) = 4(3)x passes through (2, 36).
- Another function, f(x) = 5(4)x, represents an exponential function through the point (2, 80).
Properties of Exponential Functions
- General truths about exponential functions include a domain of all real numbers.
- Inputs in these functions correspond to exponents, while the base indicates the rate of change in value.
Interest Accumulation Example
- Josiah's investment of $360 at an annual interest of 3% can be modeled by the equation y = 360(1.03)x, demonstrating how investments grow over time.
Rate of Change in Exponential Functions
- The multiplicative rate of change for functions such as f(x) = 2 * 5x is consistently 5.
- Different functions may exhibit their own rates of change, exemplified by a table that can communicate this information.
Visual Representation of Initial Values
- When graphing, plotting the initial value is crucial, as illustrated by the function f(x) = 3(4)x.
Summary of Function Behavior
- Understanding the underlying properties and equations of exponential functions is crucial for predicting growth in populations and assets.
- The consistency of certain mathematical characteristics, like domain and range, aids in the identification and analysis of these functions.
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Description
Explore the characteristics of exponential growth functions, including their graphical representation, population growth examples, and the appreciation of collectibles. This quiz will test your understanding of exponential growth concepts and their applications in real-life scenarios.
