To determine which graph represents [tex]\( g(x) = (x-2)^2 - 3 \)[/tex], we need to understand how the transformations affect [tex]\( f(x) = x^2 \)[/tex].
1. Horizontal Shift: The term [tex]\((x-2)\)[/tex] indicates a horizontal shift. Specifically, it translates the graph 2 units to the right. This means that every point on the graph of [tex]\( f(x) \)[/tex] moves 2 units right to form [tex]\( g(x) \)[/tex].
2. Vertical Shift: The term [tex]\(-3\)[/tex] at the end of the function [tex]\( g(x) \)[/tex] indicates a vertical shift downward by 3 units. This means that every point on the graph of [tex]\( f(x) \)[/tex] moves 3 units down to form [tex]\( g(x) \)[/tex].
3. Combined Shifts: Combining these transformations, the graph of [tex]\( f(x) \)[/tex] is shifted 2 units to the right and 3 units down to obtain the graph of [tex]\( g(x) \)[/tex].
Now, let’s verify this by checking some points. We compare some points [tex]\((x, f(x))\)[/tex] with the corresponding points [tex]\((x, g(x))\)[/tex]:
- For [tex]\(x = -1\)[/tex]:
- [tex]\( f(-1) = (-1)^2 = 1 \)[/tex]
- [tex]\( g(-1) = (-1-2)^2 - 3 = (-3)^2 - 3 = 9 - 3 = 6 \)[/tex]
- For [tex]\(x = 0\)[/tex]:
- [tex]\( f(0) = 0^2 = 0 \)[/tex]
- [tex]\( g(0) = (0-2)^2 - 3 = (-2)^2 - 3 = 4 - 3 = 1 \)[/tex]
- For [tex]\(x = 1\)[/tex]:
- [tex]\( f(1) = 1^2 = 1 \)[/tex]
- [tex]\( g(1) = (1-2)^2 - 3 = (-1)^2 - 3 = 1 - 3 = -2 \)[/tex]
- For [tex]\(x = 2\)[/tex]:
- [tex]\( f(2) = 2^2 = 4 \)[/tex]
- [tex]\( g(2) = (2-2)^2 - 3 = 0^2 - 3 = 0 - 3 = -3 \)[/tex]
- For [tex]\(x = 3\)[/tex]:
- [tex]\( f(3) = 3^2 = 9 \)[/tex]
- [tex]\( g(3) = (3-2)^2 - 3 = 1^2 - 3 = 1 - 3 = -2 \)[/tex]
- For [tex]\(x = 4\)[/tex]:
- [tex]\( f(4) = 4^2 = 16 \)[/tex]
- [tex]\( g(4) = (4-2)^2 - 3 = 2^2 - 3 = 4 - 3 = 1 \)[/tex]
The points we get from [tex]\( f(x) \)[/tex] are: [tex]\( ( -1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16) \)[/tex].
With the transformations applied, the points for [tex]\( g(x) \)[/tex] are: [tex]\( ( -1, 6), (0, 1), (1, -2), (2, -3), (3, -2), (4, 1) \)[/tex].
These points should help us identify the graph of [tex]\( g(x) = (x-2)^2 - 3 \)[/tex]. The graph we are looking for will have the characteristics of being the graph of [tex]\( f(x) = x^2 \)[/tex] shifted 2 units to the right and 3 units down.
