Lesson 9: Transformations of Rational Functions

A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form $$f(x) = \frac{P(x)}{Q(x)}$$ where P and Q are polynomials.

The parent function of rational functions is $$f(x) = \frac{1}{x}$$ .

Translating a function means shifting its graph horizontally, vertically, or both without changing its shape.

A vertical translation shifts the graph of a function up or down. For example, $$f(x) + k$$ shifts the graph up by k units if k > 0 and down if k < 0.

A horizontal translation shifts the graph of a function left or right. For example, $$f(x - h)$$ shifts the graph right by h units if h > 0 and left if h < 0.

Reflecting a function over the x-axis means that for every point (x, y) on the graph, the point (x, -y) will also be on the graph.

Reflecting a function over the y-axis means that for every point (x, y) on the graph, the point (-x, y) will also be on the graph.

This transformation shifts the graph of the parent function $$f(x) = \frac{1}{x}$$ up by 2 units.

This transformation shifts the graph of the parent function $$f(x) = \frac{1}{x}$$ to the right by 3 units.

This transformation reflects the graph of the parent function $$f(x) = \frac{1}{x}$$ over the x-axis.