Parent Functions and Transformations of Linear/Absolute Function

A parent function is the simplest form of a function in a family of functions. It serves as a reference point for transformations.

The parent function of absolute value is $$f(x) = |x|$$ .

Shifting a graph to the right means adding a positive value to the x-variable in the function, resulting in a horizontal translation.

Multiplying a function by a positive constant greater than 1 vertically stretches the graph away from the x-axis.

A negative sign in front of an absolute value function reflects the graph vertically across the x-axis.

A vertical shift is represented by adding or subtracting a constant from the function. For example, $$f(x) + k$$ shifts the graph up by k units, while $$f(x) - k$$ shifts it down by k units.

The function $$f(x) = |x| + 3$$ is the parent absolute value function shifted up 3 units.

Reflecting a graph vertically means flipping it over the x-axis, which is done by multiplying the function by -1.

The equation for an absolute value function shifted right 5 units is $$f(x) = |x - 5|$$ .

The function $$f(x) = 2|x + 4| - 1$$ is stretched vertically by a factor of 2, shifted left 4 units, and shifted down 1 unit.