Let's analyze the parabola given by the equation:
\[ f(x) = -4(x + 5)^2 + 6 \]
Characteristics of the Parabola
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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: \((-5, 6)\)
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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.
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Axis of Symmetry: The axis of symmetry can be found using the \( x = h \) form. So, the axis of symmetry is:
Axis of symmetry: \( x = -5 \)
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Domain: The domain of a parabolic function is all real numbers unless specified otherwise.
Domain: \( (-\infty, \infty) \)
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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: \( (-\infty, 6] \)