To determine the translation of the graph from [tex]\( y = 2(x-15)^2 + 3 \)[/tex] to [tex]\( y = 2(x-11)^2 + 3 \)[/tex], we need to examine the changes in the equation, particularly within the parentheses. Here's a step-by-step explanation:
1. Original Equation: [tex]\( y = 2(x - 15)^2 + 3 \)[/tex]
- The vertex form of this quadratic function indicates a horizontal shift based on the term [tex]\( (x - 15) \)[/tex].
- The graph has a vertex at [tex]\( (15, 3) \)[/tex].
2. New Equation: [tex]\( y = 2(x - 11)^2 + 3 \)[/tex]
- The vertex form here shows a new horizontal term [tex]\( (x - 11) \)[/tex].
- The graph has a new vertex at [tex]\( (11, 3) \)[/tex].
3. Comparison of Horizontal Shifts:
- In the original equation, the graph is shifted 15 units to the right from the origin because of [tex]\( x - 15 \)[/tex].
- In the new equation, the graph is shifted 11 units to the right from the origin because of [tex]\( x - 11 \)[/tex].
4. Determining the Translation:
- The shift from [tex]\( (x - 15) \)[/tex] to [tex]\( (x - 11) \)[/tex] indicates a movement to the right.
- Specifically, to move from [tex]\( x - 15 \)[/tex] to [tex]\( x - 11 \)[/tex], we see that the shift is [tex]\( 15 - 11 = 4 \)[/tex] units.
Therefore, the translation from the graph [tex]\( y = 2(x-15)^2 + 3 \)[/tex] to [tex]\( y = 2(x-11)^2 + 3 \)[/tex] is best described as a horizontal shift 4 units to the right.
