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.
Write an absolute-value equation from a vertex, opening direction, and arm slope
Problem
Write the absolute-value equation for the graph with vertex \((2,-3)\), upward opening, and arm slope magnitude \(1\). Start with \(y=a|x-h|+k\), determine \(a\), \(h\), and \(k\) from the features, and verify a point \(\text{one}~\text{unit}\) from the vertex on the graph scaffold.
Absolute-value vertex form assigns a separate job to each parameter. We’ll use the vertex for the horizontal and vertical shifts, use arm-slope magnitude and opening direction for the signed scale, and verify the resulting rule at a nearby point.
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