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 effect of multiplying a function by a scalar greater than 1?

Back

Multiplying a function by a scalar greater than 1 results in a vertical stretch of the graph.

Front

What does it mean to shift a graph vertically?

Back

Shifting a graph vertically means moving it up or down along the y-axis without changing its shape.

Front

What is the transformation represented by g(x) = -f(x)?

Back

The transformation represented by g(x) = -f(x) is a reflection of the graph of f across the x-axis.

Front

How does the equation g(x) = f(x) + k affect the graph of f?

Back

The equation g(x) = f(x) + k shifts the graph of f vertically by k units; if k is positive, it shifts up, and if k is negative, it shifts down.

Front

What is the effect of a negative scalar on a function?

Back

A negative scalar reflects the graph of the function across the x-axis and scales it vertically.

Front

What does the equation g(x) = f(ax) represent when a > 1?

Back

The equation g(x) = f(ax) represents a horizontal compression of the graph of f by a factor of 1/a.

Front

What is a vertical translation of a graph?

Back

A vertical translation of a graph is a shift of the entire graph up or down along the y-axis.

Front

What does it mean for a function to be steep?

Back

A function is considered steep if it has a large absolute value of the slope, indicating a rapid change in y for a small change in x.

Front

How do you determine the new equation of a function after a vertical shift?

Back

To determine the new equation after a vertical shift, add or subtract the number of units shifted to the original function's output.