- The function \(f(x) = \log_2 x + 5\) is obtained by adding a positive constant \(k=5\) to the original function \(g(x) = \log_2 x\).
- Adding a positive constant to a function results in a vertical shift upwards.
- Specifically, for every value of \(x\), the value of \(f(x)\) is 5 greater than \(g(x)\).
- The correct transformation is to \(\boxed{\text{Translate each point of the graph of } g(x) \text{ 5 units up.}}\)
Explanation
- Analyzing the Relationship Between the Functions
The functions given are \(g(x) = \log_2 x\) and \(f(x) = \log_2 x + 5\). To find the relationship between these two functions, we observe the difference: \(f(x) - g(x) = (\log_2 x + 5) - \log_2 x = 5\). This indicates that for any input \(x\), the value of \(f(x)\) is exactly 5 units greater than the value of \(g(x)\).
- Understanding Vertical Translation Rules
In coordinate geometry, shifting a function \(y = h(x)\) vertically is achieved by adding a constant \(k\) to the function: \(y = h(x) + k\). If \(k > 0\), the entire graph of the function shifts upwards by \(k\) units. If \(k < 0\), the graph shifts downwards by \(|k|\) units. Here, \(f(x) = g(x) + 5\), which means we have a constant \(k = 5\). Since \(k\) is positive, this corresponds to a vertical translation.
- Conclusion
Based on the rule identified, adding 5 to the function \(g(x)\) means that every point \((x, y)\) on the graph of \(g(x)\) is transformed into a point \((x, y + 5)\) on the graph of \(f(x)\). This geometric operation is equivalent to translating the entire graph of \(g(x)\) upward by 5 units along the vertical y-axis. Therefore, the correct transformation is to translate each point of the graph of \(g(x)\) 5 units up.
Examples
Imagine you are adjusting the volume settings on a sound system. The base volume curve \(g(x)\) represents the loudness at different frequencies. If you turn the master volume dial up by 5 decibels, you are applying a vertical shift to the entire loudness curve. Every frequency's output is increased by the same amount, just like shifting the logarithmic graph upward. Understanding these transformations is essential for engineers, designers, and anyone working with systems where input-output levels need to be scaled or shifted uniformly.