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Problem 655
655
In the diagram, the graph of \(y = f(x)\) is shown with solid lines. The graph shown with broken lines is a transformation of \(y = f(x)\).
(a) Describe fully the two single transformations of \(y = f(x)\) that have been combined to give the resulting transformation.
(b) State in terms of \(y, f\) and \(x\), the equation of the graph shown with broken lines.
Solution
(a) The transformation involves two steps:
1. A stretch parallel to the \(y\)-axis by a factor of 3. This means every \(y\)-coordinate of the graph \(y = f(x)\) is multiplied by 3.
2. A translation 4 units in the positive \(x\)-direction. This shifts the graph horizontally to the right by 4 units.
(b) The equation of the transformed graph can be written as \(y = 3f(x-4)\). This accounts for the stretch by a factor of 3 and the horizontal translation by 4 units.