Graphs of polynomials

1
Graphs of polynomials
degree n
angt 0
anlt 0
degree 3
degree 4
n even
n odd
f(x)x32x2-x-2
g(x)-x45x2-4
h(x)anxnan-1xn-1a1xa0
2 turning points
3 turning points
At most (n-1) turning points
2
When you zoom out, the effect of the lower degree
terms is difficult to see
f(x)x32x2-x-2
Close up
Zoom out
looks like graph of cubic m(x)x3
3
Theorem for approximating zeros of polynomials

• Given a polynomial P(x)anxnan-1xn-1a1xa0
• If r is a zero of the polynomial then
• r lt Max an-1/an, a1/an, a0 /an

4
Using the theorem to select a graphing window
Polynomial f(x)-2x3-3x29x10
View graph between the x-values -5 and 5
Max3/2, 9/2, 10/2 5 so for any zero r, r
lt5 Thus a zero of this function must lie between
-5 and 5.