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:

1 answer

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