To fully describe the translation from the function [tex]\( y = x^2 \)[/tex] to the function [tex]\( y = x^2 + 4 \)[/tex], we need to analyze how the graph of the function is shifted.
1. Original Function: The original function is [tex]\( y = x^2 \)[/tex]. This is a parabola with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Translated Function: The translated function is [tex]\( y = x^2 + 4 \)[/tex]. This is a parabola that has the same shape as the original, but it is shifted vertically.
3. Determining the Direction of Translation: The translation of a graph in the vertical direction is represented by adding or subtracting a constant to the original function. In this case, the term [tex]\( +4 \)[/tex] indicates that each point on the graph of [tex]\( y = x^2 \)[/tex] has been moved 4 units upward along the y-axis.
4. Translation Vector:
- A translation vector is expressed in the form [tex]\( (a, b) \)[/tex], where [tex]\( a \)[/tex] is the horizontal shift and [tex]\( b \)[/tex] is the vertical shift.
- Here, the graph has not shifted horizontally, so [tex]\( a = 0 \)[/tex].
- The vertical shift is 4 units upwards, so [tex]\( b = 4 \)[/tex].
5. Vector Form of the Translation: Combining these, the translation vector is given by [tex]\( (0, 4) \)[/tex].
Therefore, the translation from [tex]\( y = x^2 \)[/tex] to [tex]\( y = x^2 + 4 \)[/tex] is described by the vector [tex]\( (0, 4) \)[/tex].
