Question

1. f(x) = -2(x + 5)^2 - 3

Explanation:

Assuming the problem is to analyze the quadratic function \( f(x) = -2(x + 5)^2 - 3 \) (e.g., find vertex, direction of opening, etc.):

Step1: Recall vertex form of quadratic

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
For \( f(x) = -2(x + 5)^2 - 3 \), rewrite \( x + 5 \) as \( x - (-5) \), so \( h = -5 \), \( k = -3 \), \( a = -2 \).

Step2: Determine vertex

From the form, vertex is \((h, k) = (-5, -3)\).

Step3: Determine direction of opening

Since \( a = -2 < 0 \), the parabola opens downward.

Step4: (Optional) Analyze transformations

• Parent function: \( y = x^2 \) (opens up, vertex at (0,0)).
• Horizontal shift: 5 units left (due to \( x + 5 \)).
• Vertical stretch: by factor of 2 (due to \( |a| = 2 \)).
• Reflection: over x - axis (due to \( a < 0 \)).
• Vertical shift: 3 units down (due to \( -3 \)).

Answer:

• Vertex: \((-5, -3)\)
• Direction of opening: Downward
• Transformations: Shift 5 left, stretch by 2, reflect over x - axis, shift 3 down (depending on the specific question asked, e.g., if asking for vertex, the answer is \((-5, -3)\); if asking for direction, it's downward, etc.)

If the specific question was, for example, "Find the vertex of \( f(x) = -2(x + 5)^2 - 3 \)", the answer is \(\boldsymbol{(-5, -3)}\).