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 adding a constant to a function, such as f(x) + c?

Back

Adding a constant c to a function f(x) results in a vertical translation of the graph of f by c units. If c is positive, the graph shifts up; if c is negative, it shifts down.

Front

How does the transformation g(x) = f(x) + 3 affect the graph of f(x)?

Back

The transformation g(x) = f(x) + 3 shifts the graph of f(x) vertically upward by 3 units.

Front

What does the transformation g(x) = -f(x) represent?

Back

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

Front

What is the result of a vertical translation of a graph?

Back

A vertical translation of a graph moves the entire graph up or down without changing its shape.

Front

What is the effect of multiplying a function by -1, such as g(x) = -f(x)?

Back

Multiplying a function by -1 reflects the graph of the function across the x-axis.

Front

What does a horizontal translation of a graph involve?

Back

A horizontal translation involves shifting the graph left or right, which is achieved by adding or subtracting a constant from the input variable x.

Front

If f(x) is transformed to g(x) = f(x - 2), what type of transformation occurs?

Back

The transformation g(x) = f(x - 2) represents a horizontal translation of the graph of f(x) to the right by 2 units.

Front

What is the effect of a vertical stretch on a graph?

Back

A vertical stretch occurs when the output of a function is multiplied by a factor greater than 1, making the graph taller and steeper.

Front

What is the effect of a vertical compression on a graph?

Back

A vertical compression occurs when the output of a function is multiplied by a factor between 0 and 1, making the graph shorter and wider.