Graphing Absolute Value Functions

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

The graph of $$y = |x|$$ is a V-shaped graph that opens upwards, with its vertex at the origin (0,0). It is symmetric about the y-axis.

The graph of $$y = |x| + 2$$ is the same as $$y = |x|$$ but shifted upwards by 2 units.

The negative sign in $$y = -|x|$$ reflects the graph of $$y = |x|$$ across the x-axis, resulting in an upside-down V shape.

The coefficient 'a' in $$y = a|x|$$ affects the steepness of the graph. If |a| > 1, the graph is steeper; if 0 < |a| < 1, the graph is wider.

The vertex of the function $$y = |x - 2|$$ is at the point (2, 0). This is where the graph changes direction.

To find the x-intercepts, set $$|x + 3| - 1 = 0$$ . This gives $$|x + 3| = 1$$ , leading to two equations: $$x + 3 = 1$$ and $$x + 3 = -1$$ .

Adding a constant 'c' to an absolute value function shifts the graph vertically. If c > 0, it shifts up; if c < 0, it shifts down.

The domain of the function $$y = |x|$$ is all real numbers, expressed as @@(- ext{∞}, ext{∞})@@.

The range of the function $$y = |x|$$ is all non-negative real numbers, expressed as $$[0, \text{∞})$$ .