ABS transformations

An absolute value function is a function that returns the non-negative value of a number. It is defined as $$f(x) = |x|$$ , where |x| is the absolute value of x.

The transformation 'Down 5' means that the entire graph of the function is shifted down by 5 units. For example, if the original function is $$f(x)$$ , the transformed function is $$f(x) - 5$$ .

The domain of the absolute value function $$f(x) = |x|$$ is all real numbers, which means any value of x can be input into the function.

The transformation 'Up 5' means that the entire graph of the function is shifted up by 5 units. For example, if the original function is $$f(x)$$ , the transformed function is $$f(x) + 5$$ .

The range of the absolute value function $$f(x) = |x|$$ is all non-negative real numbers, which means $$y \geq 0$$ .

The range of the negative absolute value function $$f(x) = -|x|$$ is all non-positive real numbers, which means $$y \leq 0$$ .

The transformation 'Left 3' means that the entire graph of the function is shifted left by 3 units. For example, if the original function is $$f(x)$$ , the transformed function is $$f(x + 3)$$ .

The transformation 'Right 3' means that the entire graph of the function is shifted right by 3 units. For example, if the original function is $$f(x)$$ , the transformed function is $$f(x - 3)$$ .

A transformation in mathematics refers to a change in the position, size, or shape of a graph or function. Common transformations include translations, reflections, stretches, and compressions.

Multiplying an absolute value function by a negative number reflects the graph across the x-axis. For example, $$f(x) = -|x|$$ reflects the graph of $$f(x) = |x|$$ .