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 slope of a line in the equation y = mx + b?

Back

The slope (m) represents the steepness of the line, indicating how much y changes for a unit change in x.

Front

How does the transformation f(x) = kf(x) affect the graph of f(x)?

Back

If k > 1, it results in a vertical stretch; if 0 < k < 1, it results in a vertical compression.

Front

What does it mean to shift a function vertically?

Back

Shifting a function vertically means moving the graph up or down without changing its shape.

Front

What is the effect of adding a constant to a function, such as f(x) + c?

Back

Adding a constant c shifts the graph of the function vertically by c units.

Front

How does the transformation g(x) = f(x - h) affect the graph of f(x)?

Back

This transformation shifts the graph of f(x) horizontally to the right by h units if h > 0 and to the left if h < 0.

Front

What is the parent function for linear equations?

Back

The parent function for linear equations is f(x) = x, which represents a straight line with a slope of 1.

Front

What does it mean for a line to be steeper?

Back

A line is steeper if it has a greater absolute value of slope, meaning it rises or falls more sharply.

Front

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

Back

Multiplying by a negative number reflects the graph across the x-axis.

Front

What is the significance of the y-intercept in a linear equation?

Back

The y-intercept is the point where the line crosses the y-axis, represented by the value of b in the equation y = mx + b.