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

How does the graph of a function change when it is shifted left by 'k' units?

Back

The graph of the function moves left by 'k' units, while the slope remains the same.

Front

What effect does multiplying a function by a scalar greater than 1 have on its graph?

Back

It makes the graph steeper, while the y-intercept remains the same.

Front

What does the transformation h(x) = f(x) - 3 represent?

Back

It represents a vertical shift of the graph downwards by 3 units, with the slope remaining unchanged.

Front

What is the effect of the transformation h(x) = f(x + k)?

Back

It shifts the graph of the function to the left by 'k' units.

Front

What happens to the y-intercept when a function is multiplied by a scalar?

Back

The y-intercept remains the same regardless of the scalar multiplication.

Front

What is the general form of a linear transformation in two dimensions?

Back

The general form is T(x, y) = (ax + by, cx + dy), where a, b, c, and d are constants.

Front

How do you determine the slope of a transformed linear function?

Back

The slope of a transformed linear function is determined by the coefficient of x in the function's equation.

Front

What does it mean for a transformation to be 'steeper'?

Back

A transformation is 'steeper' when the absolute value of the slope increases, making the graph rise or fall more sharply.

Front

What is the relationship between the transformations g(x) = kf(x) and the steepness of the graph?

Back

If k > 1, the graph becomes steeper; if 0 < k < 1, the graph becomes less steep.