Answer:
See graph in attachment
Step-by-step explanation:
We want to graph the function,
[tex]g(x)=-2x^2-12x-24[/tex]
First, let us rewrite the function in the vertex form;
[tex]g(x)=-2(x^2+6x)-24[/tex]
[tex]\Rightarrow g(x)=-2(x^2+6x+(3)^2)--2(-3)^2-24[/tex]
[tex]\Rightarrow g(x)=-2(x^2+6x+(3)^2)+2(9)-24[/tex]
[tex]\Rightarrow g(x)=-2(x+3)^2+18-24[/tex]
[tex]\Rightarrow g(x)=-2(x+3)^2-6[/tex]
The parabola opens downwards because [tex]a=-2\:<\:0[/tex]
The vertex of the parabola is [tex](-3,-6)[/tex].
At y-intercept, [tex]x=0[/tex].
This implies that,
[tex]g(0)=-2(0+3)^2-6=-18-6=-24[/tex]
At x-intercept, [tex]y=0[/tex]
This implies that;
[tex]\Rightarrow 0=-2(x+3)^2-6[/tex]
[tex]\Rightarrow -2(x+3)^2=6[/tex]
[tex]\Rightarrow (x+3)^2=-3[/tex]
This equation has no real number solutions because of [tex]-3[/tex] on the right hand side. This implies that the graph has no x-intercepts.
We therefore draw a maximum graph through the vertex and the y-intercept to obtain the graph in the attachment.
