AP Precalculus Study Guide: Dimensions of Modeling & Transformations

Transformation Formula

The general formula for transforming a function: g(x) = a ⋅ f(b(x - h)) + k.

Vertical Stretch

Occurs when |a| > 1 in the transformation formula.

Vertical Compression

Occurs when 0 < |a| < 1 in the transformation formula.

Reflection over the x-axis

Occurs when a < 0 in the transformation formula.

Horizontal Compression

Occurs when |b| > 1 in the transformation formula.

Horizontal Stretch

Occurs when 0 < |b| < 1 in the transformation formula.

Reflection over the y-axis

Occurs when b < 0 in the transformation formula.

Horizontal Shift

Moving a graph left or right through the parameter h in the transformation.

Vertical Shift

Moving a graph up or down through the parameter k in the transformation.

Order of Transformations

The standard order is: Horizontal Shift, Horizontal Operations, Vertical Operations, Vertical Shift.

Correlation Coefficient (r)

Measures the strength and direction of a linear relationship.

Coefficient of Determination (r²)

Represents the proportion of variation in the dependent variable explained by the independent variable.

Residual

The vertical distance between the actual data point and the predicted model value.

Random Scatter in Residual Plot

Indicates that the chosen model is appropriate.

Patterned Shape in Residual Plot

Indicates that the chosen model is inappropriate.

First Differences

Used to identify linear models based on constant differences in output values.

Second Differences

Used to determine quadratic models where the first differences are not constant.

Concave Up

Indicates that the rate of change is increasing.

Concave Down

Indicates that the rate of change is decreasing.

Piecewise Functions

Functions defined by different formulas over distinct intervals of the domain.

Continuity at the Break

A function is continuous at a 'break point' if the values match from both sides.

Composition of Functions

Using the output of one function as the input for another, denoted as (f ∘ g)(x) = f(g(x)).

Domain Constraint in Composition

The domain of f(g(x)) consists of all x for which g(x) is in the domain of f.

Factor-Out Failure

Mistake of not factoring out constants before applying horizontal shifts.

Order of Operations in Function Transformations

PEMDAS applies to function mapping; perform stretches/reflections before shifts.

Residual Misinterpretation

Patterns in residual plots indicate the model missed underlying behaviors.

Composition Domains

Must consider the restrictions of the inner function g(x) before simplification.

Piecewise Continuity Check

Evaluate both rules at the break point to check for continuity.