To understand the transformation that occurs from the graph of [tex]\( f(x) = x^2 \)[/tex] to [tex]\( g(x) = (x+3)^2 + 4 \)[/tex], we need to break down the changes in the function.
1. Starting with [tex]\( f(x) = x^2 \)[/tex]:
This is the original function, a standard parabola that opens upward with its vertex at the origin [tex]\((0,0)\)[/tex].
2. Transformation to [tex]\( g(x) = (x+3)^2 \)[/tex]:
The term [tex]\((x+3)\)[/tex] inside the square indicates a horizontal shift. In general, [tex]\((x - h)\)[/tex] would shift the graph horizontally to the right by [tex]\(h\)[/tex] units. Conversely, [tex]\((x + h)\)[/tex] shifts the graph to the left by [tex]\(h\)[/tex] units. Here, [tex]\((x + 3)\)[/tex] means we shift the graph to the left by 3 units.
So, the first part of our transformation is a left shift by 3 units.
3. Transformation to [tex]\( g(x) = (x+3)^2 + 4 \)[/tex]:
Adding 4 outside the square [tex]\((x+3)^2\)[/tex] indicates a vertical shift. In general, adding a positive number [tex]\(k\)[/tex] to the function [tex]\(f(x)\)[/tex] shifts the graph upward by [tex]\(k\)[/tex] units, while subtracting [tex]\(k\)[/tex] shifts it downward. Here, adding 4 means we shift the graph upward by 4 units.
So, the second part of our transformation is an up shift by 4 units.
Combining these two transformations:
- Move the graph of [tex]\( f(x) = x^2 \)[/tex] to the left by 3 units.
- Then move it upward by 4 units.
Therefore, the transformation from the graph of [tex]\( f(x) = x^2 \)[/tex] to [tex]\( g(x) = (x+3)^2 + 4 \)[/tex] can best be described as:
left 3, up 4
So the answer is: left 3, up 4.
