parabolas

 

The vertex of the parabola described by the equation $y=-3x^2-30x-81+15x-9$ is $(m,n)$. What is $n$?   

 

                                   y  =  –3x² – 30x – 81 + 15x – 9    

 

                                   y  =  –3x² – 15x – 90    

 

The vertex is where the curve turns around and goes the other way.   

At that point, the slope of the curve is zero. 

The first derivative of an equation is the slope.   

So, set the first derivative equal to zero and solve for x.   

 

                                   y'  =  –6x – 15     

 

                                   –6x – 15  =  0          

                                           –6x  =  15         

                                               x  =  –2.5    

 

Plug x back into   

original equation           y  =  –3(–2.5²) – 15(–2.5) – 90   

 

                                      y  =  –3(6.25) + 37.5 – 90     

 

                                      y  =  –18.75 + 37.5 – 90   

 

                                      y  =  –71.25     (the "n" asked for in the problem is the y-coordinate)  

_.