Question

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

Explanation:

Assuming the problem is to identify the vertex of the parabola \( f(x) = -2(x + 5)^2 - 3 \) and possibly graph it, here's the solution:

Step 1: Recall the vertex form of a parabola

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

Step 2: Identify \( h \) and \( k \) from the given function

In \( f(x) = -2(x + 5)^2 - 3 \), we can rewrite \( (x + 5) \) as \( (x - (-5)) \). So, comparing with \( a(x - h)^2 + k \), we have \( h = -5 \) and \( k = -3 \).

Step 3: Determine the vertex

Thus, the vertex of the parabola is at \( (-5, -3) \).

To graph it:

• The coefficient \( a = -2 \) is negative, so the parabola opens downward.
• The vertex is at \( (-5, -3) \). We can plot this point.
• To find other points, we can choose \( x \)-values around \( x = -5 \), plug them into the function to find corresponding \( y \)-values, and plot those points, then draw the parabola opening downward from the vertex.

Answer:

The vertex of the parabola \( f(x) = -2(x + 5)^2 - 3 \) is \((-5, -3)\), and the parabola opens downward with this vertex.