- The transformation is identified as a horizontal shift of \(h = -5\) and a vertical shift of \(k = 2\).
- The original point \((0,0)\) is translated by the rule \((x+h, y+k)\).
- Substituting the values gives \((0-5, 0+2)\).
- The new location of the point is \(\boxed{(-5, 2)}\).
Explanation
- Problem Analysis and Translation Rule
The function \(f(x) = x^3\) has a characteristic point at the origin \((0,0)\). We are given a translated function \(g(x) = (x+5)^3 + 2\). A translation of a function \(f(x)\) to a function of the form \(g(x) = f(x - h) + k\) represents a horizontal shift of \(h\) units and a vertical shift of \(k\) units. The point \((x, y)\) on the original function \(f\) maps to a new point \((x', y') = (x + h, y + k)\) on the translated function \(g\). Our task is to determine the values of \(h\) and \(k\) for the given transformation and apply them to the origin.
- Identifying Translation Parameters
To identify the translation parameters \(h\) and \(k\), we express \(g(x)\) in the form \((x - h)^3 + k\). Comparing \(g(x) = (x+5)^3 + 2\) to \((x - h)^3 + k\), we can see that:
\[(x - h) = (x + 5) \implies h = -5\]
\[k = 2\]
This means the graph has been shifted 5 units to the left (horizontal translation of -5) and 2 units upward (vertical translation of +2).
- Calculating the New Coordinates
Now, we apply the translation transformation to the point \((0,0)\) on the graph of \(f(x)\). Using the formula \((x', y') = (x + h, y + k)\) with \(x = 0, y = 0, h = -5,\) and \(k = 2\), we calculate the new coordinates:
\[x' = x + h = 0 + (-5) = -5\]
\[y' = y + k = 0 + 2 = 2\]
Thus, the point \((0,0)\) on the original function is now located at \((-5, 2)\) on the translated function.
Examples
This concept is vital in fields like data analysis and engineering, where we often shift datasets or signal waves to align them for comparison. For example, if you have a sensor tracking a movement pattern \(f(x)\) starting at time zero, and a later event shifts the start of that movement pattern to a different time or amplitude, you can use these coordinate transformations to map where your starting data point has relocated in the new system. Mastering these transformations allows you to quickly adjust and synchronize your mathematical models to match real-world observations.