The functions f(x) = –(x − 1)2 + 5 and g(x) = (x + 2)2 − 3 have been rewritten using the completing-the-square method. Is the vertex for each function a minimum or a maximum? Explain your reasoning for each function.

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Answer 1 · 2018-04-12T21:01:18

If we write a quadratic in vertex form:

#y=a(x-h)^2+k#

Then:

#bbacolor(white)(8888)# is the coefficient of #x^2#

#bbhcolor(white)(8888)# is the axis of symmetry.

#bbkcolor(white)(8888)# is the max/min value of the function.

Also:

If #a>0# then the parabola will be of the form #uuu# and will have a minimum value.

If #a<0# then the parabola will be of the form #nnn# and will have a maximum value.

For the given functions:

#a<0#

#f(x)=-(x-1)^2+5color(white)(8888)# this has a maximum value of #bb5#

#a>0#

#f(x)=(x+2)^2-3 color(white)(8888888)# this has a minimum value of #bb(-3)#