Linear Transformations

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

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

Shifting a graph down by 'k' units results in subtracting 'k' from the function's output, changing f(x) to f(x) - k.

The transformation g(x) = -f(x) reflects the graph of f across the x-axis.

The graph of f has been shifted down 4 units.

The parent function of linear transformations is f(x) = x.

The transformation g(x) = f(x) - 6 represents a vertical shift of the graph down by 6 units.

The new equation is f(x) = x - 7.

The general form of a linear transformation can be expressed as T(x) = Ax + b, where A is a matrix and b is a vector.

The slope indicates the rate of change of the function; it represents how much y changes for a unit change in x.