Sure, let's go through the detailed steps to understand how the graph of [tex]\( f(x) = x^2 \)[/tex] is translated to form [tex]\( g(x) = (x-2)^2 - 3 \)[/tex]:
### Step-by-Step Solution
1. Understand the Original Graph [tex]\( f(x) = x^2 \)[/tex]:
- [tex]\( f(x) = x^2 \)[/tex] is a basic parabola opening upwards with its vertex at the origin [tex]\((0, 0)\)[/tex].
- The standard shape and properties of this parabola are well known.
2. Horizontal Translation:
- The expression [tex]\( (x-2)^2 \)[/tex] indicates a horizontal shift.
- Specifically, [tex]\( (x-2) \)[/tex] inside the square means we shift the graph 2 units to the right.
- This is because replacing [tex]\( x \)[/tex] by [tex]\( x-2 \)[/tex] translates each point by 2 units in the positive [tex]\( x \)[/tex]-direction.
3. Vertical Translation:
- The expression [tex]\( -3 \)[/tex] outside the square indicates a vertical shift.
- Subtracting 3 means we shift the graph downwards by 3 units.
- This is because the [tex]\( -3 \)[/tex] translates each point of the graph 3 units downwards.
4. Combining Both Translations:
- First, take the original graph [tex]\( f(x) = x^2 \)[/tex] and shift it 2 units to the right. The vertex (which was at [tex]\((0, 0)\)[/tex]) moves to [tex]\((2, 0)\)[/tex].
- Then, shift this new graph 3 units downward. The vertex that was at [tex]\((2, 0)\)[/tex] now moves to [tex]\((2, -3)\)[/tex].
- The new vertex of the translated graph [tex]\( g(x) \)[/tex] is at [tex]\((2, -3)\)[/tex].
5. Equation of the Translated Graph:
- After performing these translations, the equation of the new graph is [tex]\( g(x) = (x-2)^2 - 3 \)[/tex].
### Conclusion
The graph [tex]\( g(x) = (x-2)^2 - 3 \)[/tex] represents a parabola that:
- Opens upwards, similar to the original parabola.
- Has been shifted 2 units to the right and 3 units downwards.
- Has its vertex at [tex]\((2, -3)\)[/tex].
In essence, you should look for a graph that shows a standard parabolic curve opening upwards with its vertex at the point [tex]\((2, -3)\)[/tex].
