To understand which graph represents [tex]\( g(x) = (x-2)^2 - 3 \)[/tex], let's analyze the transformation of the basic quadratic function [tex]\( f(x) = x^2 \)[/tex].
### Step-by-Step Solution
1. Starting with the Basic Function [tex]\( f(x) = x^2 \)[/tex]:
This is a simple parabola that opens upwards with its vertex at the origin [tex]\((0,0)\)[/tex].
2. Transformation to Form [tex]\( g(x) = (x-2)^2 - 3 \)[/tex]:
The function [tex]\( g(x) \)[/tex] is derived from [tex]\( f(x) \)[/tex] by applying two transformations:
- Horizontal Translation: [tex]\( f(x) \)[/tex] is translated 2 units to the right.
- Vertical Translation: [tex]\( f(x) \)[/tex] is translated 3 units down.
#### Horizontal Translation
The expression [tex]\( (x-2)^2 \)[/tex] indicates a shift to the right by 2 units. If we replace [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex], we shift the graph of [tex]\( f(x) \)[/tex] to the right by 2 units. Hence, the vertex moves from [tex]\((0,0)\)[/tex] to [tex]\((2, 0)\)[/tex].
#### Vertical Translation
The subtraction of 3 in [tex]\( (x-2)^2 - 3 \)[/tex] indicates a shift downward by 3 units. Therefore, from the vertex [tex]\((2,0)\)[/tex] due to the horizontal shift, we now move down 3 units, resulting in the vertex at [tex]\((2, -3)\)[/tex].
### Summary of Transformations
- The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right by 2 units and down by 3 units.
- The new vertex of [tex]\( g(x) = (x-2)^2 - 3 \)[/tex] is at [tex]\((2, -3)\)[/tex].
### Characteristics of [tex]\( g(x) \)[/tex]
- The vertex is [tex]\((2, -3)\)[/tex].
- The parabola opens upwards.
- It is a standard parabola with the same "width" as the original [tex]\( f(x) \)[/tex] since there is no vertical stretching or compression.
### Graph Representation of [tex]\( g(x) \)[/tex]
The graph of [tex]\( g(x) \)[/tex] is a parabola:
- Vertex: [tex]\((2, -3)\)[/tex]
- Open upwards
- Axis of symmetry: [tex]\( x = 2 \)[/tex]
- Passes through the points derived from the transformation described (notably hitting y-intercept and easier points to calculate if needed).
### Visual Representation (Without the code)
You would see the parabola [tex]\( f(x) = x^2 \)[/tex] translated to the right by 2 units and down by 3 units. The vertex of [tex]\( g(x) \)[/tex] lies at [tex]\( (2, -3) \)[/tex], and the shape and direction (opening upwards) remain consistent with [tex]\( f(x) \)[/tex].
In conclusion, the graph of [tex]\( g(x) = (x-2)^2 - 3 \)[/tex] is the parabola [tex]\( f(x) = x^2 \)[/tex] translated right by 2 units and down by 3 units, with its vertex at [tex]\((2, -3)\)[/tex] and opening upwards.
