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For the quadratic function [tex]f(x) = x^2 + 2x + 1[/tex], answer parts (a) through (f).

(a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.

The vertex is [tex]\square[/tex] (Type an ordered pair, using integers or fractions.)

Sagot :

GinnyAnswer
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]
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