All Flashcards
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.
How do you solve an exponential equation?
- Isolate the exponential term. 2. Take the logarithm of both sides (using the same base as the exponential term). 3. Solve for the variable.
How do you solve a logarithmic equation?
- Isolate the logarithmic term. 2. Convert the equation to exponential form. 3. Solve for the variable. 4. Check for extraneous solutions.
How do you find the inverse of a function?
- Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with .
How do you evaluate a composite function?
- Evaluate the inner function first. 2. Substitute the result into the outer function. 3. Simplify.
How do you determine if two functions are inverses?
Show that and .
How do you graph an exponential function?
- Identify the initial value and growth/decay factor. 2. Plot key points. 3. Draw the curve, considering the asymptote.
How do you graph a logarithmic function?
- Identify the base. 2. Find the vertical asymptote. 3. Plot key points. 4. Draw the curve, considering the asymptote.
How do you use properties of logarithms to simplify expressions?
Apply product rule, quotient rule, and power rule to combine or separate logarithmic terms.
How to determine the equation of an exponential function from data points?
- Start with . 2. Use two data points to create a system of equations. 3. Solve for a and b.
How do you solve for the half-life in an exponential decay problem?
- Set up the equation , where b is the decay factor and t is the half-life. 2. Solve for t using logarithms.
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