To determine how to graph [tex]\( m(x) = \log_6 (x + 3) \)[/tex] based on the graph of [tex]\( h(x) = \log_6 (x) \)[/tex], let's walk through the understanding of what each transformation involves.
### Step-by-Step Explanation
1. Understanding the original function [tex]\( h(x) = \log_6 (x) \)[/tex]:
- This is a logarithmic function with base 6.
- The graph of [tex]\( h(x) \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex] (the y-axis) and passes through the point [tex]\( (1, 0) \)[/tex] because [tex]\(\log_6 1 = 0\)[/tex].
2. Understanding the transformation: [tex]\( m(x) = \log_6 (x + 3) \)[/tex]:
- The argument inside the logarithm, [tex]\( x + 3 \)[/tex], indicates a horizontal transformation.
- To understand how [tex]\( x + 3 \)[/tex] affects the graph, consider the impact on the x-values:
- If [tex]\( y = \log_6 (x) \)[/tex], then for [tex]\( y = \log_6 (x + 3) \)[/tex], the input to the logarithm is shifted.
3. How transformation works:
- The transformation [tex]\( x + 3 \)[/tex] in the function [tex]\( m(x) = \log_6 (x + 3) \)[/tex] shifts the graph horizontally.
- Specifically, for each x-value in [tex]\( h(x) \)[/tex], [tex]\( x \)[/tex] should now be replaced with [tex]\( x - 3 \)[/tex] to get [tex]\( x + 3 \)[/tex]. This means every point on the graph of [tex]\( h(x) \)[/tex] will shift to the left by 3 units.
### Conclusion
Thus, the transformation [tex]\( \log_6 (x + 3) \)[/tex] translates each point on the graph of [tex]\( h(x) = \log_6 (x) \)[/tex] to the left by 3 units. This can be stated as:
- Translate each point of the graph of [tex]\( h(x) \)[/tex] 3 units left.
So, the correct choice is:
- Translate each point of the graph of [tex]\( h(x) \)[/tex] 3 units left.
