Find a+b+c if the graph of the equation y=ax^2+bx+c is a parabola with vertex (5,3), vertical axis of symmetry, and contains the

One general formula for a vertex is:  y - k  =  a(x - h)²     <--->     This parabola has its vertex at (h,k).

 

If a parabola has a vertex (5,3), its equation can be:  y - 3  =  a(x - 5)²

To find the value of a, place (2,0) into this equation, replacing x with 2 and y with 0:   0 - 3  =  a(2 - 5)²

and solve this for a:     0 - 3  =  a(2 - 5)²     --->     -3  =  a(-3)²     --->     -3  =  a·9     --->    a  =  -1/3

 

So, the equation is:  y - 3  =  (-1/3)(x - 5)²

 

Multiplying this out:     y - 3  =  (-1/3)(x² - 10x + 25)

                                   y - 3  =  (-1/3)x² + (10/3)x - (25/3)

                                        y  =  (-1/3)x² + (10/3)x - (16/3)

 

--->     a = -1/3     b = 10/3     c = -16/3

--->     a + b + c  =  -7/3