how to graph polynomial functions

This is a part of guide to graphing functions.

• how to graph 3 degree polynomials

y-intercept

to find the y-intercept of a polynomial $f(x)$, we just need to make $x=0$ and find $f(0)$

x-intercepts

the x-intercepts of a polynomial is also called the roots of the polynomial

to find the x-intercepts (zeros) of a polynomial $f(x)$, we need to find all $x$ values for which $f(x)=0$
the most common way to do this is to factorize and use the zero-product property

s a quantity that depends on the coefficients and allows deducing some properties of the roots

end behavior

to find the end behavior of a polynomial, we can examine the leading term of the polynomial in standard form, which is a monomial, and

$f(x)=(3x-2)(x+2{)}^2$
$f(x)=(3x-2)(x^2+4x+4)$
$f(x)=3x^3+12x^2+12x-2x^2-8x-8$
$f(x)=3x^3+10x^2+4x-8$

since the leading term of the polynomial is $3x^3$,

worked example: graph $y=(x-1)(x+2)(x-3)$

x-intercepts: (1,0), (-2,0), (3,0)

the degree is 3 and the product of the x coefficients is positive, so the graph starts from bottom left and goes to upper right

worked example: graph $y=(x-1)(x+1)$
x-intercepts: (1,0), (-1,0)

the degree is 2 and the product of the x coefficients is positive, so the graph starts from upper left and goes to upper right