Linear Function Transformations

A linear function is a function that can be graphically represented as a straight line. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

The slope of a linear function represents the rate of change of the function. It indicates how much the output (y-value) changes for a unit change in the input (x-value).

Changing the slope of a linear function affects the steepness of the graph. A larger slope value results in a steeper line, while a smaller slope value results in a less steep line.

The y-intercept of a linear function is the point where the graph intersects the y-axis. It represents the value of the function when x = 0.

The graph of f(x) = x shifts to the right by c units to create the graph of g(x) = f(x - c).

The graph of f(x) = x shifts to the left by c units to create the graph of g(x) = f(x + c).

When the output of a linear function is multiplied by a fraction less than 1, the graph becomes less steep.

When the output of a linear function is multiplied by a fraction greater than 1, the graph becomes steeper.

The transformation g(x) = f(x) + k shifts the graph of f(x) vertically. If k is positive, the graph shifts up; if k is negative, the graph shifts down.

The transformation g(x) = f(x) - k shifts the graph of f(x) down by k units.