- Identify the parent function as \(f(x) = \frac{1}{x}\).
- Observe that \(g(x) = \frac{1}{x-5} + 2\) corresponds to the transformation \(f(x - h) + k\), where \(h = 5\) and \(k = 2\).
- A horizontal shift of \(h = 5\) units to the right occurs due to \(x - 5\).
- A vertical shift of \(k = 2\) units upward occurs due to the additive constant \(+2\).
- Final answer is \(\boxed{g(x) \text{ is shifted 5 units right and 2 units up from } f(x)}\).
Explanation
- Analysis of the Parent and Transformed Functions
The parent function is given by \(f(x) = \frac{1}{x}\). We are comparing it to the function \(g(x) = \frac{1}{x-5} + 2\). Transformations of a function \(f(x)\) in the form \(g(x) = f(x - h) + k\) represent a horizontal shift of \(h\) units and a vertical shift of \(k\) units. If \(h > 0\), the graph shifts right; if \(h < 0\), it shifts left. If \(k > 0\), the graph shifts up; if \(k < 0\), it shifts down.
- Determining the Transformations
To determine the transformation from \(f(x) = \frac{1}{x}\) to \(g(x) = \frac{1}{x-5} + 2\), we match the terms. We observe that \(g(x)\) has the form \(\frac{1}{x-h} + k\), where \(h = 5\) and \(k = 2\). Since \(h = 5\), which is greater than 0, the graph is shifted 5 units to the right along the horizontal axis. Since \(k = 2\), which is also greater than 0, the graph is shifted 2 units upward along the vertical axis.
- Conclusion
Comparing the derived transformation of a 5-unit shift to the right and a 2-unit shift upward with the provided options, we conclude that the correct description is that the graph of \(g(x)\) is shifted 5 units right and 2 units up from the graph of the parent function \(f(x)\). The final answer is the second option.
Examples
Understanding function transformations is essential in engineering and physics, where you might need to model waves or signals. For instance, if you have a base signal represented by a function like \(f(t) = \frac{1}{t}\), shifting it horizontally (representing time delay) or vertically (representing a bias adjustment) allows you to model different operational states of a system. Mastering these basic shifts helps you quickly visualize how changes in parameters modify the behavior of a system.