Use the graph of $y=f(x)$ to graph the function $g(x)=f(x)-3$.
Choose the correct graph of $g$ below.
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To graph the function \(g(x) = f(x) - 3\), we start by analyzing the relationship between \(f(x)\) and \(g(x)\). A vertical shift of a function \(f(x)\) is defined by the transformation \(g(x) = f(x) + k\). When \(k > 0\), the graph of \(f(x)\) is shifted upward by \(k\) units. When \(k < 0\), the graph of \(f(x)\) is shifted downward by \(|k|\) units. In our case, \(g(x) = f(x) - 3\), which corresponds to \(k = -3\). This indicates that the graph of \(f(x)\) is shifted downward by 3 units.
To accurately graph the transformation, identify key points on the original graph \(y = f(x)\), such as intercepts, local maxima, minima, or endpoints of line segments. For any point \((x, y)\) on the original graph \(y = f(x)\), the corresponding point on the transformed graph \(g(x) = f(x) - 3\) will be \((x, y - 3)\). This means we subtract 3 from the y-coordinate of every point on the graph while keeping the x-coordinate the same.
By applying this shift of 3 units downward to all parts of the original curve, we obtain the new graph for \(g(x)\). Since every vertical segment and point of the graph is moved down by exactly 3 units, the shape and orientation of the function remain unchanged; only its position relative to the x-axis is modified. The correct graph is the one that follows the original pattern but is located 3 units lower.
Vertical shifts are common in real-world scenarios. For example, if a company tracks its daily revenue \(f(x)\) throughout the week and then incurs a fixed daily expense (like a rental fee) of 3 units, the new net profit function \(g(x)\) becomes \(f(x) - 3\). By shifting the revenue graph down by the amount of the fixed expense, the business owner can easily visualize the net profit at any point in time. This transformation allows for quick adjustments in planning and financial analysis.