To analyze the function \( f(x) = \log_3(x) + 12 \), we want to determine how it has been transformed from its parent function, which is \( f(x) = \log_3(x) \).
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The Logarithmic Parent Function: The parent function \( f(x) = \log_3(x) \) has a vertical asymptote at \( x = 0 \) and passes through the point \( (1, 0) \) since \( \log_3(1) = 0 \).
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Analyzing \( f(x) = \log_3(x) + 12 \):
- The addition of \( 12 \) to the function means that the entire graph of the parent function shifts vertically upwards by \( 12 \) units.
- It does not affect the \( x \) values or the horizontal position of the graph.
Now let’s go through the options provided:
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Option #1: horizontal shrink with a factor of 13 - This is not correct because there is no \( x \) value being multiplied by a factor that would indicate a horizontal shrink.
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Option #2: vertical shift up 12 units - This is the correct transformation, as explained above.
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Option #3: horizontal shift left 12 units - This is also incorrect because there is nothing in the function that shifts it horizontally.
Therefore, the correct transformation of \( f(x) = \log_3(x) + 12 \) from its parent function is:
Option #2: vertical shift up 12 units.