Properties of Linear Transformations

Front

What is a linear transformation?

Back

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

Front

What is the general form of a linear function?

Back

The general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept.

Front

How does changing the slope of a linear function affect its graph?

Back

Changing the slope affects the steepness of the graph; a larger slope means a steeper line.

Front

What happens to the graph of a linear function when the y-intercept is increased?

Back

Increasing the y-intercept shifts the graph upward on the coordinate plane.

Front

What does it mean for a function to be 'shifted' in the context of linear transformations?

Back

A 'shift' refers to moving the graph of a function up, down, left, or right without changing its shape.

Front

If g(x) = f(x) - 4, how does the graph of g compare to the graph of f?

Back

The graph of g is shifted 4 units down from the graph of f.

Front

What is the effect of multiplying a linear function by a fraction (e.g., 1/3)?

Back

Multiplying by a fraction compresses the graph vertically, making it less steep.

Front

What does it mean if a linear function has a negative slope?

Back

A negative slope indicates that as x increases, y decreases, resulting in a downward slant of the graph.

Front

How can you determine if one linear function is steeper than another?

Back

Compare the absolute values of their slopes; the function with the larger absolute value is steeper.

Front

What is the relationship between the slopes of parallel lines?

Back

Parallel lines have the same slope but different y-intercepts.