All Flashcards
Define a linear function.
A function of the form , where is the y-intercept and is the slope.
What is the slope of a line?
The rate of change of a linear function; steepness and direction.
Define an arithmetic sequence.
A sequence with a constant difference between consecutive terms, defined as .
What is the common difference in an arithmetic sequence?
The constant value added to each term to get the next term.
Define an exponential function.
A function of the form , where is the initial value and is the base.
What is the base of an exponential function?
The factor by which the function's value changes for each unit increase in .
Define a geometric sequence.
A sequence with a constant ratio between consecutive terms, defined as .
What is the common ratio in a geometric sequence?
The constant value by which each term is multiplied to get the next term.
What is the y-intercept?
The point where the line crosses the y-axis, where .
What does the initial value represent in an exponential function?
The starting value of the function when x=0.
How do you determine the equation of a line given two points?
- Calculate the slope using . 2. Use the point-slope form: . 3. Simplify to slope-intercept form if needed.
How do you determine if a function represents exponential growth or decay?
Examine the base in . If , it's growth. If , it's decay.
How do you find the value of an exponential function after a certain number of years?
Substitute the number of years for in the function and calculate the result.
How do you find the equation of an exponential function given two points?
- Substitute the points into to get two equations. 2. Solve for in one equation. 3. Substitute the expression for into the other equation and solve for . 4. Substitute the value of back into the equation for .
How do you solve for in an exponential decay problem?
- Set up the equation . 2. Isolate the exponential term. 3. Take the logarithm of both sides. 4. Solve for .
How to determine the common difference in an arithmetic sequence?
Subtract any term from its subsequent term: .
How to determine the common ratio in a geometric sequence?
Divide any term by its preceding term: .
How do you convert an exponential function from base to base ?
Use the identity , so becomes .
How to write the equation of a line given a point and a slope?
Use the point-slope form: , where is the given point and is the slope. Then, rearrange to slope-intercept form if needed.
How do you determine the initial value of an exponential function from a table of values?
Find the y-value when x=0. This value is the initial value in the function .
What are the differences between linear and exponential functions in terms of rate of change?
Linear: Constant rate of change (constant slope). Exponential: Proportional rate of change (constant ratio).
What are the key differences between arithmetic and geometric sequences?
Arithmetic: Constant difference between terms. Geometric: Constant ratio between terms.
Compare the general forms of linear and exponential functions.
Linear: (addition). Exponential: (multiplication).
What are the differences between using addition and multiplication in linear and exponential functions?
Linear: Constant addition of slope. Exponential: Constant multiplication by the base.
Compare the parameters needed to define linear and exponential functions.
Linear: Initial value (y-intercept) and slope. Exponential: Initial value and base.
How do the domains of sequences and functions differ?
Sequences: Usually positive integers. Functions: Can have a wider range of values, like all real numbers.
Compare how linear and exponential functions are affected by transformations such as shifts and stretches.
Linear: Shifts and stretches affect slope and y-intercept directly. Exponential: Shifts and stretches affect initial value and base, impacting growth/decay rate.
What are the similarities and differences between modeling simple interest and compound interest?
Simple Interest: Modeled by linear functions (constant addition). Compound Interest: Modeled by exponential functions (constant multiplication).
Compare the long-term behavior of linear and exponential functions.
Linear: Grows (or decays) at a constant rate. Exponential: Grows (or decays) much faster in the long run.
How do you determine if a real-world situation is best modeled by a linear or an exponential function?
Linear: Look for constant addition/subtraction. Exponential: Look for constant multiplication/division (percentage increase/decrease).