- The function \(m(x) = \log_6(x+3)\) is a transformed version of \(h(x) = \log_6(x)\).
- A transformation of the form \(f(x+k)\) with \(k > 0\) represents a horizontal shift of the graph \(k\) units to the left.
- For \(m(x) = \log_6(x+3)\), we have \(k = 3\), meaning the graph shifts 3 units left.
- Final Answer: \(\boxed{\text{Translate each point of the graph of } h(x) \text{ 3 units left.}}\)
Explanation
- Understanding Function Transformations
The function \(h(x) = \log_6(x)\) is a standard logarithmic function. We are comparing this to the function \(m(x) = \log_6(x+3)\). In coordinate geometry, when we modify a function \(f(x)\) into the form \(f(x+k)\), the resulting graph is a horizontal translation of the original. Specifically, if \(k\) is a positive constant, replacing \(x\) with \(x+k\) shifts the entire graph \(k\) units to the left, because every point \((x, y)\) on the graph of \(f(x)\) corresponds to a point \((x-k, y)\) on the graph of \(f(x+k)\).
- Analyzing the Specific Shift
Comparing the given functions, we see that \(m(x) = h(x+3)\). This indicates that the graph of \(m(x)\) is obtained by taking the graph of \(h(x)\) and applying a horizontal transformation where \(k = 3\). Since 3 is positive, this transformation corresponds to a shift of the entire graph 3 units to the left along the x-axis.
- Conclusion of the Transformation
By applying the rule for horizontal shifts, we conclude that the transformation required to obtain the graph of \(m(x) = \log_6(x+3)\) from the graph of \(h(x) = \log_6(x)\) is to shift every point of the graph of \(h(x)\) 3 units to the left. Comparing this to the provided options, this matches the second choice. Therefore, the correct action is to translate each point of the graph of \(h(x)\) 3 units left.
Examples
Consider a company tracking its growth over time where the number of users \(N\) at time \(t\) follows a logarithmic pattern \(N(t) = \log_6(t)\). If the company wants to model a scenario where they start tracking their users 3 months earlier, their new model might look like \(N(t) = \log_6(t+3)\). Understanding this transformation helps analysts visualize how a change in the starting point shifts the growth curve left on the timeline, without needing to re-calculate every single data point. This is a fundamental concept in data science and business analytics for comparing different versions of models.