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Book says approximating even function $P_3$ must be of form $ax^2+b\ $so they neglected $x,x^3\ $. This function approximate even function $|x|^3$ very nicely in the book after some tweaking.

I am thinking if I was to approximate an odd fuction with polynomial of degree 3 should I ditch term $x^2\ $ and have it as $ax^3 + bx + c\ $ When I draw this function it make sense. Here is the graph of modified polynomial and 2 odd functions $x^3\ $ and $sin x$. enter image description here

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    $\begingroup$ With the constant $c=0$ your degree-three function is odd, otherwise it is not. What is your question exactly? $\endgroup$ – abiessu Sep 15 at 3:08
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    $\begingroup$ You wouldn’t even had $c$ for odd functions. $\endgroup$ – Thomas Andrews Sep 15 at 3:10
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    $\begingroup$ Don't forget that zero is an even number and that the constant term can be thought of as the "$x^0$ term." Remember also your definition for an even function and an odd function. An even function is one such that $f(x)=f(-x)$ while an odd function is one such that $f(x)=-f(-x)$. $\endgroup$ – JMoravitz Sep 15 at 3:11
  • $\begingroup$ I don't understand how a constant can change function from odd to even ? $\endgroup$ – tt z Sep 15 at 3:12
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    $\begingroup$ Rephrased in more layman terms., an even function has reflective symmetry across the $y$ axis. An odd function has rotational symmetry of $180^\circ$ around the origin. By adding a constant to an odd function, it would still have rotational symmetry but not around the origin anymore. Note also that every odd function must satisfy that $f(0)=0$ since $f(0)=-f(-0)$. $\endgroup$ – JMoravitz Sep 15 at 3:17

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