Linear Transformations

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

The effect of a reflection across the x-axis on the function f(x) is that the output values are negated, resulting in g(x) = -f(x).

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

A graph is steeper by a factor of 'k' when the slope of the function is multiplied by 'k', resulting in a vertical stretch of the graph.

When f(x) = x is reflected across the x-axis, the new function is g(x) = -x.

The parent function of linear transformations is f(x) = x, which represents a straight line through the origin with a slope of 1.

When a function is shifted up by 3 units, its equation changes from f(x) to f(x) + 3.

A reflection in the graph of a function can be identified by observing that the output values are negated, indicating a flip over the x-axis or y-axis.

The transformation represented by g(x) = 2f(x) is a vertical stretch of the graph of f(x) by a factor of 2.

A function is reflected across the y-axis when the output values are negated for all input values, resulting in g(x) = f(-x).