Linear Transformations (Translations)

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 a translation in the context of linear transformations?

Back

A translation is a type of linear transformation that shifts a graph or function horizontally or vertically without changing its shape.

Front

How does the function g(x) = f(x) + 5 transform f(x)?

Back

This transformation shifts the graph of f(x) upward by 5 units.

Front

How does the function g(x) = f(x - 3) transform f(x)?

Back

This transformation shifts the graph of f(x) to the right by 3 units.

Front

What effect does g(x) = f(x) - 2 have on the graph of f(x)?

Back

This transformation shifts the graph of f(x) downward by 2 units.

Front

If g(x) = f(x + 4), what is the transformation of f(x)?

Back

The graph of f(x) is shifted to the left by 4 units.

Front

What is the general form of a translation transformation?

Back

The general form is g(x) = f(x - h) + k, where h is the horizontal shift and k is the vertical shift.

Front

What does a positive value of k indicate in the transformation g(x) = f(x) + k?

Back

A positive value of k indicates a vertical shift upward.

Front

What does a negative value of h indicate in the transformation g(x) = f(x - h)?

Back

A negative value of h indicates a shift to the right.

Front

How can you identify a translation from a function's equation?

Back

Look for changes in the input (x) or output (f(x)) that indicate shifts, such as adding or subtracting constants.