All Flashcards
What are the differences between arithmetic and geometric sequences?
Arithmetic: Constant difference, linear growth | Geometric: Constant ratio, exponential growth
What are the differences between exponential growth and exponential decay?
Growth: b > 1, increasing function | Decay: 0 < b < 1, decreasing function
What are the differences between exponential and logarithmic functions?
Exponential: Rapid growth/decay, horizontal asymptote | Logarithmic: Slow growth, vertical asymptote
What are the differences between and ?
: Inverse function | : Reciprocal of the function
What are the differences between linear and logarithmic scales?
Linear: Equal intervals represent equal changes | Logarithmic: Equal intervals represent equal proportional changes
Compare and contrast composite functions and inverse functions.
Composite: Combining functions sequentially | Inverse: Undoing a function
Compare the domain and range of exponential and logarithmic functions.
Exponential: Domain: all reals, Range: positive reals | Logarithmic: Domain: positive reals, Range: all reals
Compare solving exponential and logarithmic equations.
Exponential: Isolate, take logarithm | Logarithmic: Isolate, convert to exponential
Compare graphing exponential and logarithmic functions.
Exponential: Horizontal asymptote, rapid growth/decay | Logarithmic: Vertical asymptote, slow growth
Compare the effect of base on exponential and logarithmic functions.
Exponential: Larger base = faster growth/decay | Logarithmic: Base influences steepness and position
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.
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.