Question

Identify the characteristics of the parabola with the given equation below:

f, of, x, equals, 3, left parenthesis, x, plus, 2, right parenthesis, squared, minus, 1f(x)=3(x+2)
2
−1

Answer
Attempt 1 out of 3

What is the vertex?


Is the vertex a maximum point or a minimum point?


What is the axis of symmetry?


Domain:


Range:

Answer 1

To analyze the given equation of the parabola \( f(x) = 3(x + 2)^2 - 1 \), we will identify its characteristics:

### 1. Vertex
The general form of a parabola in vertex form is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.

In this case:
- \( a = 3 \)
- \( h = -2 \) (note the sign change)
- \( k = -1 \)

**Vertex**: \((-2, -1)\)

### 2. Maximum or Minimum Point
Since \( a = 3 \) (which is positive), the parabola opens upwards. Therefore, the vertex represents a minimum point.

**Minimum Point**: Yes, the vertex is a minimum point.

### 3. Axis of Symmetry
The axis of symmetry can be found using the \( x \)-coordinate of the vertex. The equation of the axis of symmetry is \( x = h \).

**Axis of Symmetry**: \( x = -2 \)

### 4. Domain
The domain of a parabola is all real numbers unless otherwise restricted.

**Domain**: \( (-\infty, \infty) \)

### 5. Range
Since the parabola opens upwards and has its minimum point at the vertex \( (-2, -1) \), the \( y \)-values start from \(-1\) and go to \(\infty\).

**Range**: \( [-1, \infty) \)

### Summary
- **Vertex**: \((-2, -1)\)
- **Minimum Point**: Yes
- **Axis of Symmetry**: \( x = -2 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [-1, \infty) \)