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Questions and Answers
What does the variable 'b' represent in the exponential function model y = a * bx?
What does the variable 'b' represent in the exponential function model y = a * bx?
If b < 1 in the exponential function, it represents exponential growth.
If b < 1 in the exponential function, it represents exponential growth.
False (B)
Write the equation for a population of 200 that is decaying at a rate of 10% per year.
Write the equation for a population of 200 that is decaying at a rate of 10% per year.
f(x) = 200 * 0.9^x
Exponential decay can be modeled using the function y = a * b^x where ________ is less than 1.
Exponential decay can be modeled using the function y = a * b^x where ________ is less than 1.
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Which of the following transformations represents a vertical shift down by 4 units?
Which of the following transformations represents a vertical shift down by 4 units?
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Match the transformations with their descriptions:
Match the transformations with their descriptions:
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The equation f(x) = 3√x represents an exponential function.
The equation f(x) = 3√x represents an exponential function.
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What is the initial value in the exponential function f(x) = 150 * 1.03^x?
What is the initial value in the exponential function f(x) = 150 * 1.03^x?
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Study Notes
Exponential Growth and Decay
- Exponential growth describes a quantity increasing over time at a constant percentage rate.
- Exponential decay describes a quantity decreasing over time at a constant percentage rate.
- Both are modeled by the function y = a * bx, where:
- 'a' is the initial value (when x = 0).
- 'b' is the growth/decay factor. If b > 1, it's growth; if 0 < b < 1, it's decay.
- 'x' is the time variable.
Writing Exponential Equations with Rational Exponents
- Rational exponents represent fractional powers. Example: x(1/2) = √x.
- Exponential functions can include rational exponents in their base or exponent. For example:
- f(x) = 2(x/2)
- f(x) = 3√x
- These functions still follow the general form of exponential growth or decay but involve non-integer exponents.
- The growth or decay factor is found by the exponent's fractional part, considering the base's fractional powers.
Transformations of Exponential Functions
- Transformations of exponential functions work similarly to transformations of other functions.
- Vertical Shifts: Adding a constant 'c' to the function shifts it vertically. f(x) + c shifts up 'c' units, and f(x) - c shifts down 'c' units.
- Horizontal Shifts: Adding or subtracting a constant 'c' inside the function's exponent shifts it horizontally. f(x-c) shifts right 'c' units, and f(x+c) shifts left 'c' units.
- Vertical Stretches/Compressions: Multiplying the function by a constant 'a' stretches or compresses it. a*f(x) is a vertical stretch if |a| > 1, and a compression if 0 < |a| < 1.
- Reflections: Multiplying the function by -1 reflects it across the x-axis (f(-x)). Reflecting across the y-axis involves changing the input x to -x.
- Combining transformations alters the function's position, shape, and orientation. For example, f(x−2) + 3 represents a horizontal shift to the right by 2 units and a vertical shift up by 3 units.
Writing Exponential Functions
- To write an exponential function, you need to identify key features:
- Initial value (a): The value of the function when the input (x) is zero.
- Growth/decay factor (b): The factor by which the output is multiplied in each unit of change in the input. This is often determined by knowing a second point on the graph.
- Time variable (x).
- Example: If an initial population of 100 grows by 5% every year, use the formula f(x) = a * bx and define a = 100, and b =1 + 0.05 = 1.05. f(x) = 100 * 1.05x.
- Often, you're given multiple points on a graph (or data) and need to solve for these factors. Using two points allows the creation of a system of equations and solving using substitution or elimination.
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Description
This quiz explores the concepts of exponential growth and decay, along with writing exponential equations using rational exponents. Understand how to model these functions and their transformations, with a focus on their mathematical representations. Test your knowledge on the principles behind exponential behavior.
