Question

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

Explanation:

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

Step 1: Recall the vertex form of a parabola

The vertex form of a parabola is given by \( y=a(x - h)^2 + k \), where \((h,k)\) is the vertex of the parabola.

Step 2: Rewrite the given function in the form to identify \( h \) and \( k \)

We have the function \( g(x)=-2(x + 5)^2-3 \). We can rewrite \( (x + 5) \) as \( (x-(-5)) \). So comparing with \( y = a(x - h)^2+k \), we have \( a=-2 \), \( h=-5 \) and \( k = - 3 \).

Step 3: Determine the vertex

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

To graph the parabola:

• The coefficient \( a=-2 \) is negative, so the parabola opens downwards.
• The vertex is at \( (-5,-3) \). We can find a few more points to plot. For example, when \( x=-4 \), \( g(-4)=-2(-4 + 5)^2-3=-2(1)^2-3=-2 - 3=-5 \). When \( x=-6 \), \( g(-6)=-2(-6 + 5)^2-3=-2(-1)^2-3=-2 - 3=-5 \). When \( x=-3 \), \( g(-3)=-2(-3 + 5)^2-3=-2(2)^2-3=-8 - 3=-11 \). When \( x=-7 \), \( g(-7)=-2(-7 + 5)^2-3=-2(-2)^2-3=-8 - 3=-11 \). Then we can plot these points \((-5,-3)\), \((-4,-5)\), \((-6,-5)\), \((-3,-11)\), \((-7,-11)\) and draw the parabola opening downwards.

Answer:

The vertex of the parabola \( g(x)=-2(x + 5)^2-3 \) is \(\boldsymbol{(-5,-3)}\) and the parabola opens downwards with the vertex at this point (and can be graphed using the additional points found).