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9709 P12 - Nov 2021 - Q2
651
The graph of \(y = f(x)\) is transformed to the graph of \(y = f(2x) - 3\).
(a) Describe fully the two single transformations that have been combined to give the resulting transformation.
(b) The point \(P(5, 6)\) lies on the transformed curve \(y = f(2x) - 3\). State the coordinates of the corresponding point on the original curve \(y = f(x)\).
Solution
(a) The transformation from \(y = f(x)\) to \(y = f(2x)\) involves a horizontal compression by a factor of \(\frac{1}{2}\). This is because the function \(f(2x)\) compresses the x-values by a factor of 2, effectively scaling the x-direction by \(\frac{1}{2}\).
The transformation from \(y = f(2x)\) to \(y = f(2x) - 3\) involves a vertical translation downwards by 3 units. This is because subtracting 3 from the function shifts the graph down by 3 units.
(b) To find the corresponding point on the original curve \(y = f(x)\), we reverse the transformations. The point \(P(5, 6)\) on \(y = f(2x) - 3\) corresponds to \(y = f(2x)\) at \((5, 9)\) after reversing the vertical translation.
Next, reverse the horizontal compression by multiplying the x-coordinate by 2, giving \((10, 9)\) on \(y = f(x)\).
