- Recognize that the target function \(g(x) = x^2 - 7\) is the original function \(f(x) = x^2\) minus a constant.
- Recall that the transformation \(y = f(x) + k\) represents a vertical shift of the graph by \(k\) units.
- Identify that with \(k = -7\), the graph shifts downwards by 7 units.
- Conclude that the correct transformation is option d. \(\boxed{d}\)
Explanation
- Analysis of the Functions
The problem asks to identify the transformation that maps the graph of the function \(f(x) = x^2\) to the graph of the function \(g(x) = x^2 - 7\). The function \(f(x) = x^2\) represents a standard parabola opening upwards with its vertex at the origin \((0, 0)\). The new function \(g(x) = x^2 - 7\) is related to \(f(x)\) by the subtraction of a constant value.
- Identifying the Transformation
In general, a transformation of the form \(y = f(x) + k\) corresponds to a vertical shift of the graph of \(y = f(x)\) by \(k\) units. If \(k > 0\), the graph is shifted upwards by \(k\) units. If \(k < 0\), the graph is shifted downwards by \(|k|\) units. Here, we compare the given functions:
\[g(x) = f(x) - 7\]
Comparing this to \(y = f(x) + k\), we find \(k = -7\). Since \(k\) is negative, the graph of \(f(x) = x^2\) is shifted downwards by \(7\) units.
- Comparison and Conclusion
We evaluate the options:
AA translation 7 units up would be \(f(x) + 7 = x^2 + 7\).
BA translation 7 units to the right would be \(f(x-7) = (x-7)^2\).
CA translation 7 units to the left would be \(f(x+7) = (x+7)^2\).
DA translation 7 units down is \(f(x) - 7 = x^2 - 7\).
Therefore, the correct transformation is a translation of 7 units down, which matches option d.
Examples
Vertical translations are commonly used in various real-life scenarios to describe changes in elevation or baseline levels. For example, if a company tracks its monthly profit using a specific curve, and suddenly every month the profit decreases by a fixed amount due to an increase in operational costs, the entire graph of the profit function shifts downwards. Understanding these transformations allows engineers, economists, and data analysts to quickly model changes in systems without having to recalculate every point on the graph.