To determine the graph of the function [tex]\( g(x) = (x-2)^2 - 3 \)[/tex] from the graph of [tex]\( f(x) = x^2 \)[/tex], we follow the process of graph transformations. Here is a step-by-step explanation of how [tex]\( f(x) \)[/tex] is modified to form [tex]\( g(x) \)[/tex]:
1. Starting with [tex]\( f(x) = x^2 \)[/tex]:
The base function [tex]\( f(x) = x^2 \)[/tex] is a basic parabola with its vertex at the origin [tex]\((0,0)\)[/tex].
2. Horizontal Translation:
The expression [tex]\( (x-2) \)[/tex] inside the squared term indicates a horizontal shift. Specifically, [tex]\( (x-2) \)[/tex] means a shift to the right by 2 units. This is because we substitute [tex]\( x \)[/tex] with [tex]\( (x-2) \)[/tex], shifting every point on the graph 2 units to the right.
3. Vertical Translation:
The [tex]\( -3 \)[/tex] outside the squared term indicates a vertical shift. Specifically, it shifts the graph downward by 3 units. After shifting the graph right by 2 units, we decrease each y-value by 3 to get the final placement.
Therefore, the graph of [tex]\( g(x) = (x-2)^2 - 3 \)[/tex] is derived from [tex]\( f(x) = x^2 \)[/tex] by shifting the entire graph 2 units to the right and 3 units down.
To summarize:
- Start with the parabola [tex]\( f(x) = x^2 \)[/tex].
- Shift the graph 2 units to the right to get the function [tex]\( (x-2)^2 \)[/tex].
- Then shift this new graph 3 units downward to obtain [tex]\( g(x) = (x-2)^2 - 3 \)[/tex].
The resulting graph of [tex]\( g(x) \)[/tex] will be a parabola opening upwards, with its vertex at the point [tex]\((2, -3)\)[/tex].
Hence, [tex]\( g(x) = (x-2)^2 - 3 \)[/tex] represents a graph that is a translated version of [tex]\( f(x) = x^2 \)[/tex] with the vertex of the parabola moved to the point [tex]\((2, -3)\)[/tex].
