Question

1. reinforce what values of a, h, and k will transform the graph of the original function, f(x), shown with the solid line, so that it matches the graph of the new function rule y = af(x - h)+k, the dashed line?

Explanation:

Step1: Analyze horizontal shift

The vertex of the original function (solid - line) is at the origin \((0,0)\) and the vertex of the new function (dashed - line) is at \((- 5,0)\). The formula for a horizontal shift is \(y = f(x - h)\), where \(h\) is the horizontal shift. If the graph shifts to the left by \(5\) units, then \(h=-5\).

Step2: Analyze vertical shift

The vertical position of the vertex of the new function compared to the original function has not changed. So, \(k = 0\).

Step3: Analyze vertical stretch or reflection

The original function has a slope of \(1\) for \(x\geq0\) and \(-1\) for \(x < 0\). The new function has a slope of \(- 1\) for \(x\geq - 5\) and \(1\) for \(x < - 5\). This indicates a reflection across the \(x\) - axis, so \(a=-1\).

Answer:

\(a=-1\), \(h = - 5\), \(k = 0\)