How do you find the axis of symmetry, and the maximum or minimum value of the function #f(x) =x^2 + 3#?

1 Answer
Tony B
Mar 21, 2016

#color(brown)("Axis if symmetry is the y-axis")#
#color(brown)("Minimum "->(x,y)=(0,3))#

The answer can be derived without any calculations!

Explanation:

#color(blue)("Determine if it is a minimum or a maximum")#

For a quadratic; if the #x^2# term is positive then the graph is of general shape #uu#. On the other hand, if the #x^2# term is negative then the graph is of general shape #nn#

Your equation has a positive #x^2# term. Thus we are looking for a minimum.

#color(brown)("The graph has a minimum value")#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Determine the axis of symmetry")#

Consider the standard form equation of #y=ax^2+bx+c#

There is no term for #bx# .

#color(brown)("thus the axis of symmetry is the y-axis")#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the coordinates of the minimum value")#

Suppose for your question there was no constant. Then the minimum would be at #(x,y)->(0,0)#

The constant of 3 lifts the graph up by 3 from this point. Thus the minimum is at:

#color(brown)("Minimum "->(x,y)=(0,3))#

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