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Find the vertex of the graph of the equation y = -2x^2 + 8x + 15 - 3x^2 + 7x - 25

 Apr 3, 2024
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Find the vertex of the graph of the equation y = -2x^2 + 8x + 15 - 3x^2 + 7x - 25   

 

Combine like terms                                        y  =  –5x2 + 15x – 10    

 

Take the first derivitive                                   y'  =  –10x + 15   

 

The first derivitive defines the slope of the   

curve at any point.  When the derivitive is   

equal to zero, that's where the parabola   

turns around and goes the other way.   

 

Set equal to zero, solve for x                          –10x + 15  =  0   

                                                                             x  =  –15 / –10  =  3 / 2  

This is the x-coordinate of the vertex    

 

Plug it in to the original equation                    y  =  –5x2 + 15x – 10    

This is to see what y is when x=3/2    

                                                                       y  =  (–5)(3/2)2 + (15)(3/2) – 10   

 

                                                                       y  =  (–5)(9/4) + (15)(3/2) – 10    

 

                                                                                –45 + 90 – 40        

Find the common denominator                       y  =  –––––––––––––  =  5 / 4         

                                                                                          4        

This is the y-coordinate of the vertex    

 

Put them together and the vertex is at (3/2 , 5/4)   or, if you prefer, (1.50 , 1.25)    

 

I think there's a formula for finding the vertex without using the first derivitive   

but I don't know what it is.  Maybe somebody else will chime in.   

.

 Apr 3, 2024

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