- Identify the transformation form \(y = f(x-h) + k\).
- Determine the horizontal shift \(h\) from \(x+4\), which is \(-4\), indicating a shift of 4 units left.
- Determine the vertical shift \(k\) from \(-6\), which is \(-6\), indicating a shift of 6 units down.
- Conclude that the graph is shifted 4 units left and 6 units down: \(\boxed{\text{g(x) is shifted 4 units left and 6 units down from f(x)}}\).
Explanation
- Analysis of the Function Transformation
The parent function is given by \(f(x) = \frac{1}{x}\). The transformed function is given by \(g(x) = \frac{1}{x+4} - 6\). A general translation of a function \(f(x)\) is expressed as \(y = f(x - h) + k\), where \(h\) represents the horizontal shift and \(k\) represents the vertical shift. We will analyze the components \(x + 4\) and \(-6\) to determine how the graph of \(g(x)\) relates to the graph of \(f(x)\).
- Determining the Horizontal Shift
To determine the horizontal shift \(h\), we look at the argument of the function, which is \(x + 4\). Comparing this to the form \((x - h)\), we have \(x + 4 = x - (-4)\). This means that \(h = -4\). A horizontal shift of \(h = -4\) indicates that the graph is shifted \(4\) units to the left.
- Determining the Vertical Shift
To determine the vertical shift \(k\), we look at the constant term added to the function, which is \(-6\). Comparing this to the form \(f(x - h) + k\), we see that \(k = -6\). A vertical shift of \(k = -6\) indicates that the graph is shifted \(6\) units downwards.
- Conclusion and Final Answer
By combining these two transformations, we conclude that the graph of \(g(x) = \frac{1}{x+4} - 6\) is obtained by shifting the graph of \(f(x) = \frac{1}{x}\) by 4 units to the left and 6 units down. Therefore, the correct statement is that \(g(x)\) is shifted \(4\) units left and \(6\) units down from \(f(x)\). \(\boxed{\text{g(x) is shifted 4 units left and 6 units down from f(x)}}\).
Examples
Understanding how functions shift is essential in fields like physics and engineering. For example, if you are modeling the path of a projectile or the cooling curve of an object in a room, the parent function might describe a standard scenario (like an object cooling from 100 degrees in a room at 0 degrees). If you then change the environment—such as moving the object to a room at 20 degrees or shifting the timing of the experiment—you are applying these exact horizontal and vertical shifts to your mathematical model to accurately predict the new outcome.