Linear transformations on Graphs

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 the equation g(x) = f(x) + c represent?

Back

This represents a vertical shift of the graph of f(x) by c units. If c is positive, the graph shifts up; if c is negative, it shifts down.

Front

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

Back

This represents a horizontal shift of the graph of f(x) by c units. If c is positive, the graph shifts to the right; if c is negative, it shifts to the left.

Front

What does the equation g(x) = k * f(x) represent?

Back

This represents a vertical scaling of the graph of f(x). If k > 1, the graph stretches; if 0 < k < 1, it compresses; if k < 0, it also reflects across the x-axis.

Front

What is the effect of a negative coefficient in g(x) = k * f(x)?

Back

A negative coefficient reflects the graph of f(x) across the x-axis.

Front

What does the equation g(x) = f(ax) represent?

Back

This represents a horizontal scaling of the graph of f(x). If a > 1, the graph compresses; if 0 < a < 1, it stretches.

Front

What is the relationship between the slopes of f(x) and g(x) in g(x) = k * f(x)?

Back

The slope of g(x) is k times the slope of f(x). If k > 1, the slope increases; if 0 < k < 1, the slope decreases.

Front

How do you determine the vertical shift in g(x) = f(x) + c?

Back

The value of c indicates the vertical shift. Positive c shifts the graph up, while negative c shifts it down.

Front

What is the effect of g(x) = f(x + c)?

Back

This shifts the graph of f(x) to the left by c units.

Front

What is the effect of g(x) = f(x - c)?

Back

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