Define a linear function.

A function of the form $f(x) = b + mx$ , where $b$ is the y-intercept and $m$ is the slope.

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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 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.

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).