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

Question

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