Define a linear function.
A function of the form $f(x) = b + mx$, where $b$ is the y-intercept and $m$ 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 $a_n = a_0 + dn$.
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 $f(x) = ab^x$, where $a$ is the initial value and $b$ 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 $x$.
Define a geometric sequence.
A sequence with a constant ratio between consecutive terms, defined as $g_n = g_0 * r^n$.
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 $x=0$.
What does the initial value represent in an exponential function?
The starting value of the function when x=0.
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: $f(x) = b + mx$ (addition). Exponential: $f(x) = ab^x$ (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).
What is the formula for a linear function?
$f(x) = b + mx$
What is the formula for an arithmetic sequence?
$a_n = a_0 + dn$
What is the point-slope form of a linear function?
$f(x) = y_i + m(x - x_i)$
What is the formula for an exponential function?
$f(x) = ab^x$
What is the formula for a geometric sequence?
$g_n = g_0 * r^n$
What is the formula for a geometric sequence with a known term?
$g_n = g_k * r^(n-k)$
What is the formula for an exponential function with a known point?
$f(x) = y_i * r^(x-x_i)$
Formula for arithmetic sequence with a known term?
$a_n = a_k + d(n-k)$
How do you calculate the slope ($m$) given two points $(x_1, y_1)$ and $(x_2, y_2)$?
$m = \frac{y_2 - y_1}{x_2 - x_1}$
What is the decay factor formula?
$(1 - r)$, where $r$ is the rate of decay.
