Linear Transformation with Key Attributes

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 does it mean to translate a graph vertically?

Back

Translating a graph vertically means shifting the entire graph up or down along the y-axis without changing its shape.

Front

What is the effect of adding a constant to a function f(x)?

Back

Adding a constant to a function f(x) results in a vertical translation of the graph. For example, f(x) + c shifts the graph up by c units if c > 0 and down by c units if c < 0.

Front

What is a horizontal translation of a graph?

Back

A horizontal translation shifts the graph left or right along the x-axis. For example, f(x - c) shifts the graph to the right by c units, while f(x + c) shifts it to the left by c units.

Front

What is a vertical stretch of 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. For example, f(x) becomes k*f(x) where k > 1.

Front

What happens to the graph of f(x) when it is reflected across the x-axis?

Back

Reflecting the graph of f(x) across the x-axis changes the sign of the output, resulting in -f(x). This flips the graph upside down.

Front

How does a horizontal compression affect a graph?

Back

A horizontal compression occurs when the input of a function is multiplied by a factor greater than 1, making the graph narrower. For example, f(k*x) where k > 1 compresses the graph horizontally.

Front

What is the transformation of f(x) = x when replaced with f(x - 3)?

Back

The transformation f(x - 3) represents a horizontal translation of the graph 3 units to the right.

Front

What is the transformation of f(x) = x when replaced with f(x + 5)?

Back

The transformation f(x + 5) represents a horizontal translation of the graph 5 units to the left.

Front

What does it mean for a graph to undergo a vertical translation down?

Back

A vertical translation down occurs when a constant is subtracted from the function, shifting the graph down along the y-axis.