Questions and Answers
Which graph represents Chelsea's first step in graphing the function f(x) = 20()x?
What function represents the value of Terrence's car after x years?
f(x) = 20,000(0.85)^x
Which function represents the frog population after x years?
f(x) = 1,200(0.97)^x
What is the multiplicative rate of change of the function represented in the table?
Signup and view all the answers
Which is the graph of f(x) = 3(2/3)^x?
Signup and view all the answers
What is the multiplicative rate of change of the function represented in the table?
Signup and view all the answers
Which is the graph of f(x) = 100(0.7)^x?
Signup and view all the answers
What is the multiplicative rate of change of the function represented in the table?
Signup and view all the answers
Which is an exponential decay function?
Signup and view all the answers
Which are true of the function f(x) = 49(1/7)^x? Check all that apply.
Signup and view all the answers
What conclusion can be drawn about f(x) and g(x) from the table?
Signup and view all the answers
What describes the graph of the function that represents the number of live bacteria after x days of treatment?
Signup and view all the answers
Which table shows exponential decay?
Signup and view all the answers
What multiplicative rate of change should Hal use in his function for a $10,000 investment decreasing at 2% annually?
Signup and view all the answers
What is the multiplicative rate of change of the function f(x) = 10(0.5)^x?
Signup and view all the answers
Which graph represents exponential decay?
Signup and view all the answers
Which graph represents g(x) = (1/2)^x, reflected over the y-axis?
Signup and view all the answers
Which is the graph of f(x) = 4(1/2)^x?
Signup and view all the answers
Study Notes
Exponential Functions Overview
- Exponential decay functions model scenarios where quantities decrease over time.
- Commonly expressed as f(x) = a(b)^x, where a is the initial value and b (0 < b < 1) is the decay factor.
Examples of Exponential Decay
-
Car Value Depreciation:
- Initial value: $20,000
- Depreciation rate: 15% per year
- Function: f(x) = 20,000(0.85)^x
-
Frog Population Study:
- Initial population: 1,200 frogs
- Decrease rate: 3% per year
- Function: f(x) = 1,200(0.97)^x
-
Bacteria Post-Antibiotic Treatment:
- Initial number of bacteria: 5,000
- Survivability: 40% remains after each treatment
- Function: f(x) = 5000(0.4)^x with horizontal asymptote at y = 0
Understanding Graphs of Exponential Decay
- Graphs of exponential decay show a rapid decrease in value initially, gradually approaching zero.
- Multiplicative rates of change indicate how fast the values decrease; for instance:
- 0.5 implies the function halves its value each interval.
- 0.98 indicates a 2% annual decline.
Identifying Exponential Decay Functions
- Functions like f(x) = 3/2(8/7)^-x are characterized by negative exponents, signaling decay.
- The characteristics of exponential decay include a domain of all real numbers and a range where values remain positive (y > 0).
Reflection Properties
- Exponential functions can reflect over the y-axis, which shows a switch from growth to decay or vice versa.
Additional Examples and Properties
- Table data can represent exponential decay by consistently decreasing values.
- A function like f(x) = 49(1/7)^x decreases multiplicatively, with consistent changes in y-values.
- In practical scenarios, such as investments, a rate of change less than 1 indicates decay (e.g., a 2% decrease corresponds to a factor of 0.98).
Confidential Graphs
- Identifying graph types helps understand the nature of a function; for instance, graphs depicting exponential decay curve downward, confirming the nature of the decay represented in the equations.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Learn about exponential decay functions through flashcards that provide scenarios and definitions. This quiz covers key concepts such as graphing functions and depreciation calculations. Test your understanding with various examples!
