Let's discuss the transformation of the parent function [tex]\( y = \sqrt{x} \)[/tex] to the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex].
The parent function [tex]\( y = \sqrt{x} \)[/tex] is the most basic form of the square root function, which has its vertex at the origin [tex]\((0, 0)\)[/tex].
To transform this function to [tex]\( y = \sqrt{x + 7} + 5 \)[/tex]:
1. Horizontal Shift:
- Inside the square root function, the term [tex]\(x + 7\)[/tex] indicates a horizontal shift.
- Normally, the expression inside the square root affects the function horizontally - in particular, [tex]\( y = \sqrt{x - h} \)[/tex] shifts the graph [tex]\( h \)[/tex] units to the right.
- In our case, we have [tex]\( x + 7 \)[/tex], which is equivalent to [tex]\( x - (-7) \)[/tex]. Therefore, the graph is shifted 7 units to the left.
2. Vertical Shift:
- Outside the square root function, the term [tex]\(+ 5\)[/tex] indicates a vertical shift.
- Adding a constant outside the function means it affects the graph vertically by lifting it up or down.
- Here, [tex]\(+ 5\)[/tex] added to the whole function lifts the graph 5 units up.
So, the graph of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is:
- Shifted 7 units to the left.
- Shifted 5 units up.
Therefore, the graph is shifted 7 units left and 5 units up.
