To determine which graph could represent the function [tex]\( f(x) = (x+3.8)^2 - 2.7 \)[/tex], we need to analyze its characteristics. Let's break it down step by step:
### 1. Form of the function:
The given function [tex]\( f(x) = (x+3.8)^2 - 2.7 \)[/tex] is a quadratic function, which can be written in the standard form for a parabola [tex]\( f(x) = a(x - h)^2 + k \)[/tex]. Here:
- [tex]\( a = 1 \)[/tex] (since the coefficient of the squared term is implicitly 1, which is positive),
- [tex]\( h = -3.8 \)[/tex],
- [tex]\( k = -2.7 \)[/tex].
### 2. Vertex of the parabola:
The vertex of the parabola given by [tex]\( f(x) = a(x - h)^2 + k \)[/tex] is at the point [tex]\( (h, k) \)[/tex]. For our function:
- [tex]\( h = -3.8 \)[/tex],
- [tex]\( k = -2.7 \)[/tex].
So, the vertex is [tex]\( (-3.8, -2.7) \)[/tex].
### 3. Direction of the parabola:
Since the coefficient [tex]\( a \)[/tex] of the squared term [tex]\( (x + 3.8)^2 \)[/tex] is positive (i.e., [tex]\( a = 1 \)[/tex]), the parabola opens upwards.
### 4. Overall shape and properties:
Summarizing the given function:
- The vertex of the parabola is at [tex]\( (-3.8, -2.7) \)[/tex].
- The parabola opens upwards.
Given these properties, we can confidently state that the graph representing the function [tex]\( f(x) = (x+3.8)^2 - 2.7 \)[/tex] is a parabola with its vertex located at the point [tex]\((-3.8, -2.7)\)[/tex] and it opens upwards. This defines the specific shape and location of the parabola on the coordinate plane.
