# even and odd cubic polynomials

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$$.

• With the constant $c=0$ your degree-three function is odd, otherwise it is not. What is your question exactly? – abiessu Sep 15 at 3:08
• You wouldn’t even had $c$ for odd functions. – Thomas Andrews Sep 15 at 3:10
• 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 $f(x)=-f(-x)$. – JMoravitz Sep 15 at 3:11
• I don't understand how a constant can change function from odd to even ? – tt z Sep 15 at 3:12
• 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 $f(0)=-f(-0)$. – JMoravitz Sep 15 at 3:17