Linear: Straight line, constant rate of change | Polynomial: Curve, changing rate of change

Polynomial: Can be written as a sum of terms with non-negative integer exponents. | Rational: Written as a ratio of two polynomials, may have asymptotes.

Shifts: Translate the graph without changing its shape. | Stretches: Change the shape by compressing or expanding the graph.

Direct: As one quantity increases, the other increases. | Inverse: As one quantity increases, the other decreases.

Domain: Set of possible input values (x). | Range: Set of possible output values (y).

Continuous: A single function defined over its entire domain without any breaks. | Piecewise: Defined by different functions over different intervals of its domain.

Vertical Stretch: Multiplies the y-values by a factor, making the graph taller. | Horizontal Compression: Divides the x-values by a factor, making the graph narrower.

Vertical Shift: Moves the graph up or down by adding or subtracting a constant. | Horizontal Shift: Moves the graph left or right by adding or subtracting a constant from the x-value.

Linear Regression: Finds the best-fitting straight line for the data. | Polynomial Regression: Finds the best-fitting polynomial curve for the data.

Assumptions: Simplifications made about the real-world scenario to create a tractable model. | Restrictions: Constraints on the domain or range of the function based on the real-world context.

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

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

$V = \pi r^2 h$

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

$f(x-h)$. 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.

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.