Question

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

f, of, x, equals, minus, 4, left parenthesis, x, plus, 5, right parenthesis, squared, plus, 6f(x)=−4(x+5)
2
+6

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

Let's analyze the parabola given by the equation:

\[ f(x) = -4(x + 5)^2 + 6 \]

### Characteristics of the Parabola

1. **Vertex**:
The vertex form of a parabola is generally given by \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex.
In this case, we can identify:
- \( h = -5 \)
- \( k = 6 \)

Thus, the vertex is \((-5, 6)\).

**Vertex: \((-5, 6)\)**

2. **Maximum or Minimum Point**:
Since the coefficient \( a = -4 \) is negative, the parabola opens downward. Therefore, the vertex is a maximum point.

**Vertex is a maximum point.**

3. **Axis of Symmetry**:
The axis of symmetry can be found using the \( x = h \) form. So, the axis of symmetry is:

**Axis of symmetry: \( x = -5 \)**

4. **Domain**:
The domain of a parabolic function is all real numbers unless specified otherwise.

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

5. **Range**:
As the parabola opens downward and the vertex is the maximum point at \( y = 6 \), the range of the function consists of all values less than or equal to 6.

**Range: \( (-\infty, 6] \)**

### Summary
- **Vertex**: \((-5, 6)\)
- **Maximum or Minimum**: Maximum point
- **Axis of Symmetry**: \( x = -5 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-\infty, 6] \)