How do you find the axis of symmetry, and the maximum or minimum value of the function # f(x)=-x^2+6x+6#?

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Answer 1 · 2016-02-10T11:11:36

Complete the square

The line #x=3# is a line of symmetry

The maximum is #f(3)=15#.

#f(x) = -(x-3)^2 + 15#.

Notice that

#f(3+a) = f(3-a)#

for any #a in RR#.

Therefore, the line #x=3# is a line of symmetry.

The maximum is #f(3)=15#.

This is because

#f(3)=15>=15-(x-3)^2=f(x)#

Answer 2 · 2016-02-10T11:25:03

Axis of symmetry is #x=3#.

The maximum#" "->(x,y)->(3,15)#

As the coefficient for #x^2" is "(-1)# the shape type of the quadratic is #nn# so it is a maximum.

Finding axis of symmetry: Let us look at the part of the equation that is #+6x#

Compere to the standard form #y=ax^2+bx+c#

Write standard form as: #y=a(x^2+b/ax)+c#

From this #" "x_("vertex")=(-1/2)xxb/a#

So in your case, as #" "x_("vertex")=(-1/2)xx6#

But #a=(-1)" " #so in fact what you have is:

#" "x_("vertex")=(-1/2)xx6/(-1)" "=" "+3#

Now it is just a matter of substituting #x=3# into the original equation to find that #y=15#

Tony B