Certainly! Let's analyze the function [tex]\( f(x) = (x + 3.8)^2 - 2.7 \)[/tex] and determine its graphical characteristics step-by-step.
### Step 1: Identify the Transformation
The given function [tex]\( f(x) = (x + 3.8)^2 - 2.7 \)[/tex] can be broken down as follows:
- [tex]\( (x + 3.8)^2 \)[/tex]: represents a parabola that opens upwards with its vertex shifted horizontally to [tex]\( x = -3.8 \)[/tex].
- [tex]\( -2.7 \)[/tex]: vertically translates the entire parabola down by 2.7 units.
### Step 2: Find the Vertex
The vertex form of the quadratic function [tex]\( f(x) = a(x - h)^2 + k \)[/tex] helps us identify:
- Vertex [tex]\((h, k)\)[/tex], where [tex]\( h = -3.8 \)[/tex] and [tex]\( k = -2.7 \)[/tex].
Thus, the vertex of the parabola is at [tex]\((-3.8, -2.7)\)[/tex].
### Step 3: Determine Additional Points
To further understand the shape, let's calculate the function value at several key points around the vertex:
- [tex]\( f(-3.8) = (-3.8 + 3.8)^2 - 2.7 = 0 - 2.7 = -2.7 \)[/tex]
- [tex]\( f(0) = (0 + 3.8)^2 - 2.7 = 3.8^2 - 2.7 \approx 14.44 - 2.7 = 11.74 \)[/tex]
- [tex]\( f(-7.6) = (-7.6 + 3.8)^2 - 2.7 = (-3.8)^2 - 2.7 = 14.44 - 2.7 = 11.74 \)[/tex]
These points indicate symmetry around the vertex and confirmations around typical quadratic curve behavior.
### Step 4: Sketch the Graph
Combining these details, here's how the graph would appear:
1. Vertex at [tex]\((-3.8, -2.7)\)[/tex].
2. Parabolic Shape opening upwards from the vertex.
3. Key Points include [tex]\( (-3.8, -2.7) \)[/tex], [tex]\( (0, 11.74) \)[/tex], and [tex]\( (-7.6, 11.74) \)[/tex].
### Step 5: Vertical Behavior at Extremes
To confirm the graph across the extended range [tex]\([-10, 10]\)[/tex]:
- As [tex]\( x \)[/tex] moves far from -3.8 (both positive and negative directions), [tex]\( f(x) \)[/tex] consistently increases due to the squared term's dominance.
- Values like [tex]\( x = -10 \)[/tex] and [tex]\( x = 10 \)[/tex] yield corresponding function outputs following similar analytical behaviors as calculated:
- [tex]\( f(-10) = 35.74 \)[/tex] and [tex]\( f(10) = 186.7 \)[/tex]
### Conclusion
The function [tex]\( f(x) = (x + 3.8)^2 - 2.7 \)[/tex] results in a parabolic graph with:
- Vertex at [tex]\((-3.8, -2.7)\)[/tex]
- Opening upwards
- Symmetry in points calculated previously.
Thus, the graphical representation must align with these transformations, points, and overall behavior.
In summary, the graph would show a classic U-shaped parabola shifted to the left by 3.8 units, and downwards by 2.7 units, representing all the determined values and shapes effectively.
