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Questions and Answers
What are exponential functions?
What are exponential functions?
Exponential Functions
What is the general form of the following equation: y - 2 = 3^(2x/7)?
What is the general form of the following equation: y - 2 = 3^(2x/7)?
y = 1/7 * 9^x + 4
What is the domain and range of the function with points (0, -3), (1, -2), (2, 0), (3, 4)?
What is the domain and range of the function with points (0, -3), (1, -2), (2, 0), (3, 4)?
Domain: (-inf, inf), Range: (-4, inf)
What is the initial value of the function?
What is the initial value of the function?
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What is the horizontal asymptote of the function?
What is the horizontal asymptote of the function?
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Which statement describes the end behavior of the function?
Which statement describes the end behavior of the function?
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What is the horizontal asymptote of the function f(x) = 2^x - 5?
What is the horizontal asymptote of the function f(x) = 2^x - 5?
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Which graph represents the function f(x) = 2 * 3^x + 4?
Which graph represents the function f(x) = 2 * 3^x + 4?
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Which function will have a graph that decreases?
Which function will have a graph that decreases?
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What is the domain and range of the function f(x) = 2(5)^x + 3?
What is the domain and range of the function f(x) = 2(5)^x + 3?
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What is the initial value of the function f(x) = 9(2/3)^x + 4?
What is the initial value of the function f(x) = 9(2/3)^x + 4?
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Which statements are true about the growth of certain bacteria in extreme temperatures?
Which statements are true about the growth of certain bacteria in extreme temperatures?
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What was the initial value of Rhonda's car when she bought it?
What was the initial value of Rhonda's car when she bought it?
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Is James' credit card situation an example of exponential growth or decay?
Is James' credit card situation an example of exponential growth or decay?
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What is the horizontal asymptote of the function represented by the graph link?
What is the horizontal asymptote of the function represented by the graph link?
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What is the initial value of the function indicated by the graph link?
What is the initial value of the function indicated by the graph link?
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Which function has a horizontal asymptote equal to 2?
Which function has a horizontal asymptote equal to 2?
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What is the domain and range of the function? (Select two answers: one for the domain and one for the range.)
What is the domain and range of the function? (Select two answers: one for the domain and one for the range.)
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Which graph represents the function f(x) = 4 * 3^x - 2?
Which graph represents the function f(x) = 4 * 3^x - 2?
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Which statements are true about the function f(x) = 3(0.95)^x - 5? (Select all that apply)
Which statements are true about the function f(x) = 3(0.95)^x - 5? (Select all that apply)
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What is the domain and range of the function f(x) = 3(1/3)^x - 1?
What is the domain and range of the function f(x) = 3(1/3)^x - 1?
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Which function will exhibit increasing behavior?
Which function will exhibit increasing behavior?
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What is the effect on f(x) when translated from f(x) = 2 * 3^x + 4 to g(x) = 2 * 3^x - 1?
What is the effect on f(x) when translated from f(x) = 2 * 3^x + 4 to g(x) = 2 * 3^x - 1?
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What is the transformation rule when the function is modified?
What is the transformation rule when the function is modified?
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What is the value of b in the transformation of g(x) = f(1/bx)?
What is the value of b in the transformation of g(x) = f(1/bx)?
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What is the transformation rule when g(x) = af(x)?
What is the transformation rule when g(x) = af(x)?
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What is the effect on f(x) when transformed to g(x) = 2^-x?
What is the effect on f(x) when transformed to g(x) = 2^-x?
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What is the definition of logarithms based on the equation log_2 8 = 3?
What is the definition of logarithms based on the equation log_2 8 = 3?
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What is the exponential form of log_x 109 = 12?
What is the exponential form of log_x 109 = 12?
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What is the value of log 10,000?
What is the value of log 10,000?
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What is the closest approximate value of log 3.7?
What is the closest approximate value of log 3.7?
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What is the value of log_3 81?
What is the value of log_3 81?
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Study Notes
Exponential Functions
- General form: ( y = ab^x + k )
- Example transformation: ( y - 2 = 3^{\frac{2x}{7}} ) simplifies to ( y = \frac{9}{7} * 3^x + 4 )
- Initial value is derived from function output when ( x = 0 ).
- Horizontal asymptote indicates the function's end behavior, for example, ( f(x) = 3 ) for the function approaches.
- Domain of exponential functions is all real numbers ((-∞, ∞)).
- Range of exponential functions typically starts above the horizontal asymptote, e.g., ( (-4, ∞) ).
Behavior of Functions
- As ( x ) approaches ( -∞ ), functions can stabilize towards a horizontal asymptote, while as ( x ) approaches ( ∞ ), they can tend toward ( ∞ ).
- Examples of specific functions:
- ( f(x) = 2^x - 5 ), horizontal asymptote at ( y = -5 ).
- The function ( f(x) = 1.25(0.3)^x + 7 ) is a decreasing function.
- The function ( f(x) = 2 * 3^x + 4 ) is increasing, passing through point ( (0, 6) ).
Logarithmic Functions
- Logarithms are the inverses of exponential functions, such as ( f(x) = 2^x ) having its inverse as ( f^{-1}(x) = log_2(x) ).
- Relationship between logarithmic and exponential forms: ( \log_x y = z ) means ( x^z = y ).
- Basic conversions include changing ( a^b = c ) to ( \log_a c = b ).
Evaluating Logarithms
- Common values: ( log_{10} 10,000 = 4 ), ( log_2 8 = 3 ), ( log_3 81 = 4 ).
- Approximations can be calculated, e.g., ( log 250 \approx 2.4 ), ( log 3.7 \approx 0.6 ).
- Finding the base of logarithms from equations relates to their original exponential forms.
Transformations of Exponential Functions
- Vertical translations adjust the graph up or down, while horizontal translations shift it left or right.
- Dilations modify the steepness of the graph:
- If ( g(x) = 3f(x) ), the graph is vertically stretched by a factor of 3.
- Horizontal stretches/compressions can be represented as ( g(x) = f(\frac{1}{bx}) ).
- Reflections occur when ( g(x) = -f(x) ), indicating a flip across the x-axis or y-axis based on transformation rules.
General Concepts
- Asymptotes describe the flatness of the graph on one or two sides, helping understand long-term behavior.
- Initial values give insights into where the function starts, guiding calculations for growth or decay.
- The context of function transformations can incorporate combinations of shifts, reflections, and dilations, producing diverse graphs from basic forms.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on exponential and logarithmic functions with these flashcards from Unit One of Algebra 2B. Each card covers key concepts, formulas, and examples to help you master the subject. Perfect for reviewing definitions and applications of exponential functions.
