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

Question

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