We know that the absolute value of a number is always positive (or zero).
We can see this same result reflected in the graph of the absolute value parent function y = | x |. All of the graph"s y-values will be positive (or zero).
The graph of the absolute value parent function is composed of two linear "pieces" joined together at a common vertex (the origin). The graph of such absolute value functions generally takes the shape of a V, or an up-side-down V. Notice that the graph is symmetric about the y-axis.
Linear "pieces" will appear in the equation of the absolute value function in the following manner: y = | mx + b | + c where the vertex is (-b/m, c) and the axis of symmetry is x = -b/m.
Note that the slope of the linear "pieces" are +1 on the right side and -1 on the left side. Remember that when lines are perpendicular (form a right angle) their slopes are negative reciprocals.
Features (of parent function): • Domain: All Reals (-∞,∞) Unless domain is altered. • Range: <0,∞) • increasing (0, ∞) • decreasing (-∞,0)
• positive (-∞, 0) U (0, -∞)
• absolute/relative min is 0 • no absolute max (graph → ∞)
x-intercept: intersects x-axis at (0, 0) unless transformed
y-intercept: intersects y-axis at (0, 0) unless transformed
Vertex: the point (0,0) unless transformed
Table: Y1: y = | x |
Average rate of change: is constant on each straight line section (ray) of the graph.
General Form of Absolute Value Function: f (x) = a | x - h | + k • the vertex is at (h,k) • the axis of symmetry is x = h • the graph has a vertical shift of k • the graph opens up if a > 0, down if a