Questions and Answers
What are the characteristics of an exponential growth function?
Continuous, domain is all real numbers and range positive numbers. Asymptote is the x-axis.
What is the transformation of the exponential function equation?
F(x) = ab^(x-h) + k
What does 'H' represent in the context of transformations?
Along the x-axis opposite direction.
What does 'K' represent in the context of transformations?
Signup and view all the answers
What does 'A' represent in an exponential function?
Signup and view all the answers
What are the characteristics of an exponential decay function?
Signup and view all the answers
What is the exponential decay equation?
Signup and view all the answers
What is the exponential growth function equation?
Signup and view all the answers
To solve exponential equations, get a common ______, ignore base and solve.
Signup and view all the answers
What is a logarithm?
Signup and view all the answers
The property of equality for logarithmic functions states that if log_b x = log_b y, then x = y.
Signup and view all the answers
What is the product property of logarithms?
Signup and view all the answers
What is the quotient property of logarithms?
Signup and view all the answers
What is the power property of logarithms?
Signup and view all the answers
What are common logs?
Signup and view all the answers
What is the change of base formula?
Signup and view all the answers
What is the compound equation?
Signup and view all the answers
What is the transformation of the logarithmic function equation?
Signup and view all the answers
Study Notes
Characteristics of Exponential Growth Function
- Continuous function with a domain of all real numbers.
- Range consists of positive numbers only.
- The asymptote is the x-axis.
Transformation of Exponential Function Equation
- General form: (F(x) = ab^{x-h} + k).
- Parameters (h) and (k) enable shifting the graph horizontally and vertically.
Parameter H
- Controls horizontal movement along the x-axis.
- Shifts the function in the opposite direction of its value.
Parameter K
- Governs vertical translation on the y-axis.
- Moves the graph up or down depending on its value.
Parameter A
- Defines the orientation and shape of the exponential graph.
- If (0 < |a| < 1), the function is vertically stretched.
Characteristics of Exponential Decay Function
- Also a continuous function with a domain of all real numbers.
- The range is positive numbers with an asymptote on the x-axis.
Exponential Decay Equation
- Formulated as (Y = a(1-r)^t) for decay scenarios.
Exponential Growth Function
- Expressed as (Y = a(1+r)^t), indicating growth over time.
Solving Exponential Equations
- Use a common base for both sides before ignoring the base to solve the equation.
Logarithm
- Defined as (Y = \log_b x) with (b > 0) and (b \neq 1).
- Functions as the inverse of exponential functions.
Property of Equality for Logarithmic Functions
- If (b) is a positive number (and not 1), then ( \log_b x = \log_b y) holds true if and only if (x = y).
Product Property of Logarithms
- The logarithm of a product is calculated as the sum of the logarithms of each factor: ( \log_b(xy) = \log_b x + \log_b y ).
Quotient Property of Logarithms
- The logarithm of a quotient is represented as the difference of the logarithms: ( \log_b(x/y) = \log_b x - \log_b y ).
Power Property of Logarithms
- The logarithm of a power is equal to the product of the logarithm and its exponent: ( \log_b(x^n) = n \cdot \log_b x ).
Common Logs
- Logs that have a base of 10, denoted simply as ( \log x ).
Change of Base Formula
- Enables the conversion of logarithms from one base to another: ( \log_a b = \frac{\log_n b}{\log_n a} ).
Compound Equation
- Represents the formula for compound interest: (A = P(1 + \frac{r}{n})^{nt}).
Transformation of Logarithmic Function Equation
- Formulated as (A \log_b (x-h) + k), allowing for horizontal and vertical shifts.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This set of flashcards covers key concepts from Chapter 7 of Algebra 2. Each card features important definitions and characteristics related to exponential functions and their transformations. Ideal for students looking to reinforce their understanding of this chapter's material.