To determine the values of \( y = f(13x) \) for the given points after applying a horizontal stretch by a factor of 3, we first observe that stretching horizontally by a factor of 3 affects the x-coordinates inversely. Thus, for the transformation \( f(13x) \), we evaluate where \( f(x) \) would be for the given x-values.
We can rewrite the equation as \( x = \frac{t}{13} \) where \( t \) is the x-coordinate in the original function.
Calculating the values:
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For \( x = 0 \): \[ f(13 \cdot 0) = f(0) = 0 \]
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For \( x = 3 \): \[ 13x = 13 \cdot 3 = 39 \quad \Rightarrow \quad f(39) \quad (\text{This corresponds to } t = \frac{39}{13} = 3) \quad \Rightarrow \quad f(3) = 9 \quad \Rightarrow \quad y = 1 \]
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For \( x = 4 \): \[ 13 \cdot 4 = 52 \quad \Rightarrow \quad f(52) \quad (\text{This corresponds to } t = \frac{52}{13} = 4) \quad \Rightarrow \quad f(4) = 16 \quad \Rightarrow \quad y = 16 \]
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For \( x = 9 \): \[ 13 \cdot 9 = 117 \quad \Rightarrow \quad f(117) \quad (\text{This corresponds to } t = \frac{117}{13} = 9) \quad \Rightarrow \quad f(9) = 81 \quad \Rightarrow \quad y = 81 \]
Completed Table:
\[ \begin{array}{|c|c|} \hline x & f(13x) \ \hline 0 & 0 \ 3 & 1 \ 4 & 16 \ 9 & 81 \ \hline \end{array} \]
Therefore, the completed table for \( y = f(13x) \) is:
\[ \begin{array}{|c|c|} \hline x & f(13x) \ \hline 0 & 0 \ 3 & 1 \ 4 & 16 \ 9 & 81 \ \hline \end{array} \]