- The translation occurs only in the y-direction, changing from \(-1\) to \(2\).
- Using the function \(g(y) = y + k\), we set up the equation \(2 = -1 + k\).
- Solving this equation yields \(k = 3\).
- The final answer is \(\boxed{a}\).
Explanation
- Analyze the given transformation.
The given point is translated from \((4, -1)\) to \((4, 2)\). We observe that the x-coordinate remains constant at 4, meaning there is no horizontal translation involved. The y-coordinate, however, changes from -1 to 2. This indicates a vertical shift represented by the function \(g(y) = y + k\).
- Set up the equation to solve for k.
The function \(g(y) = y + k\) defines the new y-coordinate as the result of the initial y-coordinate plus a constant \(k\). We have the initial y-coordinate \(y_{initial} = -1\) and the final y-coordinate \(y_{final} = 2\). Therefore, we can set up the equation:
\[y_{final} = y_{initial} + k\]
Substituting the known values, we get:
\[2 = -1 + k\]
- Solve for k.
To solve for \(k\), add 1 to both sides of the equation:
\[2 + 1 = -1 + 1 + k\]
\[3 = k\]
Thus, the translation constant is \(k = 3\), which corresponds to option a.
Examples
In daily life, translations are used in various fields like computer graphics and animation. For instance, if you are designing a character in a game and need to move it up the screen, you are effectively applying a vertical translation. If the character starts at a height of 2 pixels and needs to move to a height of 5 pixels, the game engine calculates the movement constant \(k\) by finding the difference: \(5 - 2 = 3\), meaning the character is translated up by 3 pixels.