Question

Given the function f(x)=log3x+12
, which of the following is the transformation of f(x)
from its parent function?

Option #1: horizontal shrink with a factor of 13

Option #2: vertical shift up 12 units

Option #3: horizontal shift left 12 units

Answer 1

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 \( f(x) = \log_3(x) \).

1. **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 \( \log_3(1) = 0 \).

2. **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:

- **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.

- **Option #2: vertical shift up 12 units** - This is the correct transformation, as explained above.

- **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.**