To determine the horizontal translation of the function \(y = f(x)\) into \(y = g(x)\), we need to compare the corresponding outputs of the two functions.
The values given in the tables for \(f(x)\) are:
\[ \begin{align*} f(-1) &= 0 \ f(0) &= 1 \ f(1) &= 2 \ f(2) &= 9 \ \end{align*} \]
The values given in the tables for \(g(x)\) are:
\[ \begin{align*} g(3) &= 0 \ g(4) &= 1 \ g(5) &= 2 \ g(6) &= 9 \ \end{align*} \]
Now, let's analyze the outputs of both functions:
- For \(f(-1) = 0\) and \(g(3) = 0\): This indicates a shift to the right by 4 units because \(-1 + 4 = 3\).
- For \(f(0) = 1\) and \(g(4) = 1\): This also confirms a shift to the right by 4 units because \(0 + 4 = 4\).
- For \(f(1) = 2\) and \(g(5) = 2\): This confirms a shift to the right by 4 units since \(1 + 4 = 5\).
- For \(f(2) = 9\) and \(g(6) = 9\): This again confirms a shift to the right by 4 units because \(2 + 4 = 6\).
In all comparisons, each \(x\) value in \(f(x)\) translates to an \(x\) value in \(g(x)\) that is 4 units greater, indicating a horizontal translation to the right.
Thus, the original function \(y = f(x)\) has been translated by 4 units to form the function \(y = g(x)\).
The original function has been translated by 4 units.
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