What is an exponential function?
A function of the form $f(x) = ab^x$, where a and b are constants, and b > 0 and not equal to 1.
What is a logarithmic function?
A function of the form $f(x) = \log_b(x)$, which is the inverse of an exponential function.
Define arithmetic sequence.
A sequence where each term differs by a constant amount (common difference).
Define geometric sequence.
A sequence where each term is multiplied by a constant amount (common ratio).
What is a composite function?
A function formed by combining two or more functions, where the output of one function becomes the input of another.
What are inverse functions?
Functions that 'undo' each other. If $f(g(x)) = x$ and $g(f(x)) = x$, then $f(x)$ and $g(x)$ are inverses.
What is a semi-log plot?
A graph that uses a logarithmic scale on one axis and a linear scale on the other.
Define the common difference in an arithmetic sequence.
The constant amount added to each term to get the next term.
Define the common ratio in a geometric sequence.
The constant amount multiplied by each term to get the next term.
What is the initial value in an exponential function?
The value of the function when x = 0 (the 'a' in $f(x) = ab^x$).
Explain the concept of exponential growth.
Exponential growth occurs when a quantity increases proportionally to its current value, leading to rapid growth over time.
Explain the concept of exponential decay.
Exponential decay occurs when a quantity decreases proportionally to its current value, leading to rapid decline over time.
Explain the relationship between exponential and logarithmic functions.
Logarithmic functions are the inverses of exponential functions. They 'undo' each other.
Explain the significance of the base 'b' in an exponential function.
If b > 1, the function represents growth. If 0 < b < 1, the function represents decay.
Explain the significance of the base 'b' in a logarithmic function.
The base determines the rate at which the logarithm increases; it also defines the domain (x > 0).
Explain the concept of asymptotes in exponential and logarithmic functions.
Exponential functions have a horizontal asymptote, while logarithmic functions have a vertical asymptote. These are lines the graph approaches but never touches.
Explain the concept of domain and range for exponential functions.
The domain is all real numbers, and the range is all positive real numbers (if a > 0).
Explain the concept of domain and range for logarithmic functions.
The domain is all positive real numbers, and the range is all real numbers.
Explain how composite functions combine the transformations of individual functions.
The inner function's output becomes the input for the outer function, applying transformations sequentially.
Explain how inverse functions reflect across the line y=x.
Graphically, a function and its inverse are reflections of each other across the line y=x.
What does the graph of an increasing exponential function tell us?
It indicates exponential growth, where the rate of increase accelerates over time.
What does the graph of a decreasing exponential function tell us?
It indicates exponential decay, where the rate of decrease slows down over time.
What does the graph of a logarithmic function tell us?
It shows a slow rate of increase, approaching a vertical asymptote.
How does the base of an exponential function affect the graph?
A larger base results in faster growth or decay.
How does the base of a logarithmic function affect the graph?
The base affects the steepness and position of the graph relative to the y-axis.
What does a semi-log plot of exponential data look like?
A straight line, indicating a constant rate of growth or decay on a logarithmic scale.
How can you determine if a graph represents an exponential function?
Look for rapid growth or decay and a horizontal asymptote.
How can you determine if a graph represents a logarithmic function?
Look for a slow rate of increase and a vertical asymptote.
What does the y-intercept of an exponential function represent?
The initial value of the function.
What does the x-intercept of a logarithmic function represent?
The value for which the argument of the logarithm is equal to 1.
