How do you find the axis of symmetry, graph and find the maximum or minimum value of the function #f(x) = -x² + 5x#?

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Answer 1 · 2017-08-27T02:59:36

Axis of symmetry: #x=2.5 forall y in RR#
#f_max = f(2.5) = 6.25#

#f(x) = -x^2+5x#

#f(x)# is a quadratic function of the form: #ax^2+bx+c#

Where: #a=-1, b=5, c=0#

The graph of #f(x)# is a parabola with axis of symmetry where #x = (-b)/(2a)#

in this case

#x= (-5)/-2 = 5/2 =2.5#

Hence, the axis of symmetry of #f(x)# is the vertical line #x=2.5 forall y in RR#

Since the coefficient of #x^2<0# #f(x)# will have a maximum value on the axis of symmetry.

#f_max = f(5/2)#

#= -(5/2)^2+5*(5/2)#

#=-(25/4) + 25/2#

#= -25/4 + 50/4 = 25/4 = 6.25#

Hence, #f_max = f(2.5) = 6.25#

Both the axis of symmetry and #f_max# can be seen on the graph of #f(x)# below.
graph{(y+x^2-5x)(x-2.5-0.000000001y)=0 [-14.24, 14.23, -7.11, 7.13]}