Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
Derive and verify absolute-value graph features
Problem
Analyze \(y=|x-3|+2\) as \(y=a|x-h|+k\). Determine the vertex, axis, opening direction, and left and right arm slopes; evaluate inputs \(\text{one}~\text{unit}\) on each side of the axis; then state the range and identify the feature set supported by every check.
Big Picture
What this problem is really about
Absolute-value form describes a V by separating its center from the slopes of its two arms. We’ll extract the vertex, axis, and opening from the parameters, test equally spaced inputs for symmetry, and use the vertex bound to state the range.
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