Function Modeling Tools for Polynomial and Rational Functions

Transformation of a function

A systematic change to a function’s equation/graph (shifts, stretches/compressions, reflections) used to build new models from a familiar base graph.

Parent function

A basic “base” function used as a starting shape for transformations (e.g., f(x)=x, x^2, x^3, 1/x).

Inside vs. outside transformations

Changes outside f( ) affect outputs (vertical changes); changes inside f( ) affect inputs (horizontal changes).

Transformation template (general form)

g(x)=a·f(b(x−h))+k, where h and b act inside (horizontal effects) and a and k act outside (vertical effects).

Horizontal shift (h)

In g(x)=f(x−h), the graph shifts right if h>0 and left if h<0.

Vertical shift (k)

In g(x)=f(x)+k, the graph shifts up if k>0 and down if k<0.

Vertical scale factor (a)

In g(x)=a·f(x), outputs are multiplied by a: |a|>1 stretches vertically, 0<|a|<1 compresses vertically, and a<0 reflects across the x-axis.

Horizontal scale factor (b)

In g(x)=f(bx), inputs are multiplied by b: |b|>1 compresses horizontally, 0<|b|<1 stretches horizontally, and b<0 reflects across the y-axis.

Vertex

A key feature of a parabola (quadratic) where the turning point occurs; under g(x)=a·f(b(x−h))+k it shifts according to (h,k).

Intercepts

Key features where a graph crosses the axes: x-intercepts occur where y=0; the y-intercept is the value at x=0 (if defined).

Zero (root)

An x-value where the function equals 0; zeros correspond to x-intercepts on the graph.

Factor form (polynomial)

A polynomial written using its zeros, e.g., P(x)=a(x−r1)(x−r2)… where each zero r gives a factor (x−r).

Scaling constant (a) in factor form

The constant multiplier in P(x)=a∏(x−ri) that controls vertical scaling (and reflection if negative); zeros alone do not determine a.

Rational function

A function of the form R(x)=P(x)/Q(x), where P and Q are polynomials and Q(x)≠0 (domain excludes values making the denominator 0).

Vertical asymptote

A vertical line x=c that the graph approaches; for g(x)=a/(x−h)+k the vertical asymptote is x=h (often where the denominator is 0 and does not cancel).

Horizontal asymptote

A horizontal line y=c that the graph approaches for large |x|; for g(x)=a/(x−h)+k the horizontal asymptote is y=k.

Hole (removable discontinuity)

A missing point in a rational function’s graph caused by a common factor canceling between numerator and denominator (not a vertical asymptote).

Function model

An equation/rule that represents a relationship between an input (independent variable) and an output (dependent variable).

Model selection (shape + context)

Choosing a function family (linear/quadratic/polynomial/rational) by matching graph shape, context meaning, and required features (turning points, intercepts, asymptotes, etc.).

Piecewise-defined function

A function defined by different formulas on different intervals of the domain; exactly one rule applies for any allowed input.

Continuity (at a boundary)

At x=c in a piecewise function, the function is continuous if the left-hand value, right-hand value, and the defined value at c all match.

Open vs. closed circle (endpoint notation)

On a piecewise graph, use an open circle when the endpoint is not included (strict inequality) and a closed circle when it is included (≤ or ≥).

Composition of functions

Combining functions so one’s output becomes the other’s input: (f∘g)(x)=f(g(x)); order matters (generally f(g(x))≠g(f(x))).

Domain of a composition

For (f∘g)(x), x must be in the domain of g and g(x) must be in the domain of f (important when g(x) creates a forbidden input for f).

Residual analysis (regression check)

In regression, a residual is r=y−ŷ (observed minus predicted). A good residual plot shows residuals scattered around 0 with no pattern; patterns suggest a different model family is needed.