- The function \(y = f(x - h) + k\) represents a translation of the graph \(f(x)\).
- Comparing \(y = f(x - 7) + 3\) with the standard form, we identify \(h = 7\) and \(k = 3\).
- A positive \(h\) signifies a horizontal shift to the right by \(7\) units.
- A positive \(k\) signifies a vertical shift upwards by \(3\) units.
- The correct description is: it is the graph of \(f\) translated \(3\) units up and \(7\) units to the right.
- The final answer is \(\boxed{\text{It is the graph of } f \text{ translated } 3 \text{ units up and } 7 \text{ units to the right.}}\)
Explanation
- Understanding Function Transformations
The given function is \(f(x) = \sqrt[3]{x}\). We are looking for the transformation of this function that results in \(y = f(x - 7) + 3\).
The general form for horizontal and vertical translations of a function \(f(x)\) is \(y = f(x - h) + k\).
- The parameter \(h\) represents a horizontal translation. If \(h > 0\), the graph is shifted \(h\) units to the right. If \(h < 0\), the graph is shifted \(|h|\) units to the left.
- The parameter \(k\) represents a vertical translation. If \(k > 0\), the graph is shifted \(k\) units up. If \(k < 0\), the graph is shifted \(|k|\) units down.
- Identifying the Shifts
Comparing the function \(y = f(x - 7) + 3\) to the general form \(y = f(x - h) + k\), we can identify the values of \(h\) and \(k\):
- For the horizontal shift, \(x - 7 = x - h\), which implies \(h = 7\). Since \(h = 7 > 0\), the graph is translated \(7\) units to the right.
- For the vertical shift, \(k = 3\). Since \(k = 3 > 0\), the graph is translated \(3\) units up.
- Conclusion
Based on our analysis, the transformation involves a horizontal shift of \(7\) units to the right and a vertical shift of \(3\) units up. Comparing this to the given options:
- It is the graph of \(f\) translated \(3\) units up and \(7\) units to the left. (Incorrect)
- It is the graph of \(f\) translated \(7\) units down and \(3\) units to the right. (Incorrect)
- It is the graph of \(f\) translated \(7\) units up and \(3\) units to the right. (Incorrect)
- It is the graph of \(f\) translated \(3\) units up and \(7\) units to the right. (Correct)
Thus, the correct statement is that the graph of \(f\) is translated \(3\) units up and \(7\) units to the right.
Examples
Understanding function transformations is crucial in fields like engineering and economics. For instance, if you are modeling the growth of a company's revenue over time, you might start with a base model \(f(t)\) and apply transformations to account for delayed start dates (horizontal shift) or initial capital injections (vertical shift). Mastering these shifts helps you predict and analyze changes in systems accurately, allowing for more precise planning and decision-making.