Restrictions, such as domain and range, ensure the model aligns with real-world constraints and produces meaningful results.

Transformations allow us to adapt basic functions to fit complex data sets by shifting, stretching, compressing, or reflecting them.

Calculators and software can perform regressions and other statistical methods to find the best-fit model for a given data set, especially for complex relationships.

They model situations where the relationship between variables changes at specific points, allowing for different functions to apply to different intervals.

In rational functions, as the denominator increases, the overall value of the function decreases, representing an inverse relationship between the variables.

Units provide context and ensure that the predictions and interpretations derived from the model are meaningful and accurate.

The domain represents the set of input values that are physically possible or meaningful in the real-world scenario being modeled. For example, time or quantity cannot be negative.

The range represents the set of output values that are physically possible or meaningful in the real-world scenario being modeled. It's the set of all possible results you can get from your model.

Assumptions simplify the real-world scenario to make it mathematically tractable. They can affect the accuracy and applicability of the model, so they must be carefully considered.

By substituting specific input values into the function, we can estimate corresponding output values, allowing us to predict future behavior or outcomes based on the model.

$F = G * \frac{m_1 * m_2}{r^2}$

$F = k_e * \frac{q_1 * q_2}{r^2}$

$V = \pi r^2 h$

$V = \frac{1}{3} \pi r^2 h$

$f(x-h)$. Right if $h > 0$, left if $h < 0$.

$f(x) + k$. Up if $k > 0$, down if $k < 0$.

$a * f(x)$. Stretch if $|a| > 1$, compress if $0 < |a| < 1$.

$-f(x)$

$f(-x)$

$f(x) = \frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials.

A mathematical representation of a real-world situation using a function.

A function defined by multiple sub-functions, each applying to a certain interval of the main function's domain.

A function that can be defined as a quotient of two polynomial functions, i.e., $f(x) = \frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials.

A relationship where one quantity decreases as another increases, often modeled by a rational function.

The set of all possible input values (x-values) for which the function is defined and makes sense in the real-world context.

The set of all possible output values (y-values) that the function can produce, considering the real-world context.

The simplest form of a function family, used as a basis for transformations. Examples: $y = x^2$, $y = x^3$, $y = \sqrt{x}$.

A change made to a parent function to fit a given data set, including shifts, stretches, compressions, and reflections.

A statistical method used to find the best-fitting linear relationship between two variables in a data set.

A measure of how one quantity changes with respect to another, often representing the slope of a function or its derivative.