To determine the function [tex]\( g(x) \)[/tex], let's start by breaking down the information given and following a step-by-step process to understand the transformation from [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex].
1. Original Function:
The given function is [tex]\( f(x) = (x + 3)^2 - 10 \)[/tex].
The vertex form of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex, helps us identify the vertex of [tex]\( f(x) \)[/tex] easily:
- Here, [tex]\((x + 3)\)[/tex] means the parabola is shifted to the left by 3 units, giving us [tex]\( h = -3 \)[/tex].
- The constant term [tex]\(-10\)[/tex] gives us [tex]\( k = -10 \)[/tex].
So, the vertex of [tex]\( f(x) \)[/tex] is at [tex]\((-3, -10)\)[/tex] and its axis of symmetry is [tex]\( x = -3 \)[/tex].
2. Translation Details:
[tex]\( g(x) \)[/tex] is stated to be a translation of [tex]\( f(x) \)[/tex] such that its axis of symmetry is 5 units to the right of the axis of symmetry of [tex]\( f(x) \)[/tex].
- If the axis of symmetry of [tex]\( f(x) \)[/tex] is [tex]\( x = -3 \)[/tex], then shifting it 5 units to the right means the new axis of symmetry is [tex]\( x = -3 + 5 = 2 \)[/tex].
This means the vertex of [tex]\( g(x) \)[/tex] should now be at [tex]\( (2, k) \)[/tex].
3. Form of the Function [tex]\( g(x) \)[/tex]:
Given [tex]\( g(x) = (x - h)^2 + k \)[/tex]:
- We have determined [tex]\( h \)[/tex] to be 2 due to the new axis of symmetry.
- The vertical shift remains unchanged because the vertical translation of the function does not alter the horizontal translation discussed.
4. Matching with Given Possible Forms:
We need to match [tex]\( g(x) = (x - 2)^2 + k \)[/tex] where [tex]\( h = 2 \)[/tex] and [tex]\( k = -10 \)[/tex]. This gives us the specific form:
- [tex]\( g(x) = (x - 2)^2 - 10 \)[/tex].
5. Comparing Options:
- The correct choice should match [tex]\( h = 2 \)[/tex] and [tex]\( k = -10 \)[/tex].
By examining the given options:
- [tex]\( g(x) = (x - 2)^2 + k \)[/tex]: This matches the form well if [tex]\( k = -10 \)[/tex].
- [tex]\( g(x) = (x + 8)^2 + k \)[/tex]: This would place [tex]\( h \)[/tex] at [tex]\(-8\)[/tex], which does not fit our conditions.
- [tex]\( g(x) = (x - h)^2 - 5 \)[/tex]: This could fit if [tex]\( h \)[/tex] is 2, but the constant term [tex]\(-5\)[/tex] does not match our vertical shift.
- [tex]\( g(x) = (x - h)^2 - 15 \)[/tex]: This could fit with [tex]\( h \)[/tex] being 2, but the constant term [tex]\(-15\)[/tex] still doesn’t match our vertical shift.
From the analysis, the correct function is [tex]\( g(x) = (x - 2)^2 - 10 \)[/tex], fitting into the form [tex]\( g(x) = (x - 2)^2 + k \)[/tex] with [tex]\( h = 2 \)[/tex] and [tex]\( k = -10 \)[/tex], which translates to the first option in the list.
