How do I use the vertex formula to determine the vertex of the graph for #f(x) = −x^2 + 18x − 75#?

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Answer 1 · 2015-05-01T16:11:59

The vertex formula has the form
#y=m(x-a)^2+b# for constants #m, a, b#
with a vertex at #(a,b)#

Converting #f(x)=-x^2+18x-75#
into the vertex formula form is a matter of "completing the square"

#(fx) = (-1)(x^2-18x+9^2) -75+9^2#

#= (-1)(x-9)^2+6#

The vertex is at #(9,6)#

Answer 2 · 2015-05-01T16:28:31

The vertex formula says that the graph of #y=ax^2+bx+c# has its vertex at the point where #x = -b/(2a)#.

In this equation, we have #a=-1# and #b=18#.

The formula tells us that the #x#-coordinate of the vertex is:

#x=-(18)/(2(-1)) = -18/-2 =-(-9)=9#

To find the #y#-coordinate, use the equation and #x=9#:

#y=-(9)^2+18(9)-75#

#y=-81+ 162-75 = 162 - 156#

#y= 6#

The vertex is #(9,6)#