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Questions and Answers
What does the variable 'C' represent in the exponential function?
What does the variable 'C' represent in the exponential function?
Which condition must be true for the base 'a' in the exponential function?
Which condition must be true for the base 'a' in the exponential function?
In what real-world applications is the exponential function primarily used?
In what real-world applications is the exponential function primarily used?
If an exponential function is defined as $f(x) = Ca^x$, which of the following represents a valid example if C = 2 and a = 3?
If an exponential function is defined as $f(x) = Ca^x$, which of the following represents a valid example if C = 2 and a = 3?
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What can be inferred if an exponential function has a base less than 1?
What can be inferred if an exponential function has a base less than 1?
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Study Notes
Exponential Functions
- Exponential functions are important in mathematics for modeling growth and decay.
- Exponential functions are used to model population growth, radioactive decay, and financial processes like interest and depreciation.
- The exponential function with base a is defined for all real numbers x by: f(x) = ax , where a > 0 and a ≠ 1.
- a is the growth factor.
- C is the initial value.
- The exponential function passes through a common point (0, 1).
- The domain is (-∞, ∞)
- The range is (0, ∞)
- The asymptote is y = 0
Exponential Growth
- f(x) = C ax where a > 1
- a is the growth factor (greater than 1)
- The function increases as x increases
Exponential Decay
- f(x) = C ax where 0 < a < 1
- a is the decay factor (fraction between 0 and 1)
- The function decreases as x increases
The Number e
- e is an irrational number approximately equal to 2.71828.
- e is important in continuous growth and decay models.
- The number e is defined as the number in the expression (1 + 1/n)n as n approaches infinity.
Graphing Exponential Functions
- Graphs of exponential functions have a characteristic shape.
- The graph always passes through (0, 1).
- The horizontal asymptote is y = 0.
- Transformations of exponential functions can be applied (e.g. shifts, reflections.)
Solving Exponential Equations
- Exponential equations involve solving for an unknown exponent.
- Techniques for solving exponential equations vary depending on the form and complexity of the equation.
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Description
Test your understanding of exponential functions, including their definitions, growth, and decay. You'll explore how these functions are applied in various real-world scenarios, including population growth and financial processes. Dive into the significance of the number e and its role in continuous growth.
