before I ask for anything I must admit I'm working hard to understand this beautiful subject. Thanks in advance. $$ f(x)= 2(x)^2+8x+5 $$ Acoording to the graph of this function, there is a x-axis symmetry. The problem is I can not prove it algebraically. Thanks again.
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Do this $$f(x) = 2(x^2 + 4x) + 5 = 2(x^2 + 4x + 4) + 5 - 8 = 2(x + 2)^2 - 3.$$ Note that $f$ is insensitive to the sign of $x + 2$ so $f$ is symmetric about the line $x = -2$
So it is symmetric on $x=-2$ which is parallel to $y$-axis.