To analyze the quadratic function [tex]\( f(x) = x^2 + 2x + 1 \)[/tex], we can follow these steps:
### Step 1: Identify the coefficients
Firstly, we need to identify the coefficients from the standard form [tex]\( f(x) = ax^2 + bx + c \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = 1 \)[/tex]
### Step 2: Find the vertex
The vertex [tex]\( (h, k) \)[/tex] can be found using the formulas:
- [tex]\( h = -\frac{b}{2a} \)[/tex]
- [tex]\( k = f(h) \)[/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ h = -\frac{2}{2 \times 1} = -1 \][/tex]
Now, substitute [tex]\( h = -1 \)[/tex] back into the function [tex]\( f(x) \)[/tex] to find [tex]\( k \)[/tex]:
[tex]\[ k = f(-1) = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0 \][/tex]
So, the vertex is:
[tex]\[ (-1, 0) \][/tex]
### Step 3: Find the axis of symmetry
The axis of symmetry is the vertical line that passes through the vertex. This is given by the line:
[tex]\[ x = h \][/tex]
Thus, the axis of symmetry is:
[tex]\[ x = -1 \][/tex]
### Step 4: Determine the concavity of the graph
The concavity of the graph is determined by the sign of the coefficient [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards (concave up).
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards (concave down).
Since [tex]\( a = 1 \)[/tex] (which is greater than 0), the graph is concave up.
### Final Answer
- The vertex is [tex]\((-1, 0)\)[/tex].
- The axis of symmetry is [tex]\( x = -1 \)[/tex].
- The graph is concave up.
Thus, for part (a), the vertex is:
[tex]\[
(-1, 0)
\][/tex]
