Question

dentify the​ vertex, the axis of​ symmetry, the maximum or minimum​ value, and the range of the parabola.yequalsxsquaredplus2xplus5Question content area bottomPart 1The vertex isenter your response here. ​(Type an ordered​ pair.)

Answer 1

To find the vertex, axis of symmetry, maximum or minimum value, and range of the parabola given by the equation \( y = x^2 + 2x + 5 \), we can follow these steps:

1. **Finding the vertex**:
The vertex form of a quadratic equation can be determined using the formula:
\[
x = -\frac{b}{2a}
\]
where \( a \) is the coefficient of \( x^2 \) and \( b \) is the coefficient of \( x \).

In the equation \( y = x^2 + 2x + 5 \):
- \( a = 1 \)
- \( b = 2 \)

Now, substituting in the values:
\[
x = -\frac{2}{2 \cdot 1} = -1
\]

Next, substitute \( x = -1 \) back into the equation to find the \( y \)-coordinate of the vertex:
\[
y = (-1)^2 + 2(-1) + 5 = 1 - 2 + 5 = 4
\]

Thus, the vertex is:
\[
\text{Vertex} = (-1, 4)
\]

2. **Axis of symmetry**:
The axis of symmetry is the vertical line that passes through the vertex. This can be given by the equation:
\[
x = -1
\]

3. **Maximum or minimum value**:
Since the coefficient of \( x^2 \) (which is \( a = 1 \)) is positive, the parabola opens upwards. Therefore, the vertex represents the minimum point of the parabola.

Minimum value of \( y \):
\[
\text{Minimum value} = 4
\]

4. **Range of the parabola**:
Since the parabola opens upwards and the minimum value is 4, the range of the function is:
\[
\text{Range} = [4, \infty)
\]

### Final Results:
- Vertex: \( (-1, 4) \)
- Axis of symmetry: \( x = -1 \)
- Minimum value: \( 4 \)
- Range: \( [4, \infty) \)