2. Smooth, Continuous Graphs
Two important features of the graphs of polynomial functions are that they are
smooth and continuous. By smooth, we mean that the graph contains only
rounded curves with no y y
sharp corners. By Smooth
rounded Smooth
rounded
continuous, we mean corner corner
that the graph has no
breaks and can be
drawn without lifting
x x
your pencil from the
rectangular coordinate
system. These ideas are Smooth Smooth
rounded
illustrated in the figure. corner
rounded
corner

3. Graphs of polynomials are smooth and continuous.
No sharp corners or cusps No gaps or holes, can be drawn
without lifting pencil from paper
This IS the graph This IS NOT the graph
of a polynomial of a polynomial

4. and
LEFT RIGHT
HAND BEHAVIOUR OF A GRAPH
The degree of the polynomial along with the sign of the
coefficient of the term with the highest power will tell us
about the left and right hand behaviour of a graph.

5. The Leading Coefficient Test
As x increases or decreases without bound, the graph of the polynomial
function
f (x) anxn an-1xn-1 an-2xn-2 … a1x a0 (an 0)
eventually rises or falls. In particular,
1. For n odd: an 0 an 0
If the If the leading
leading Rises coefficient is Rises
coefficient is right negative, the left
positive, the graph rises
graph falls to to the left
the left and and falls to
rises to the the right. Falls
right. Falls right
left

6. The Leading Coefficient Test
As x increases or decreases without bound, the graph of the polynomial
function
f (x) anxn an-1xn-1 an-2xn-2 … a1x a0 (an 0)
eventually rises or falls. In particular,
1. For n even: an 0 an 0
If the If the leading
leading Rises coefficient is
coefficient is right negative, the
positive, the Rises graph falls to
graph rises left the left and
to the left to the right.
and to the Falls
right. left
Falls
right

7. Text Example
Use the Leading Coefficient Test to determine the end behavior of the graph of
Graph the quadratic function f(x) x3 3x2 x 3.
Rises right
y
Solution Because the degree is odd
(n 3) and the leading coefficient, 1,
is positive, the graph falls to the left
and rises to the right, as shown in the
figure.
x
Falls left

8. Even degree polynomials rise on both the left and
right hand sides of the graph (like x2) if the coefficient
is positive. The additional terms may cause the
graph to have some turns near the center but will
always have the same left and right hand behaviour
determined by the highest powered term.
left hand right hand
behaviour: rises behaviour: rises

9. Even degree polynomials fall on both the left and
right hand sides of the graph (like - x2) if the
coefficient is negative.
turning points
in the middle
left hand
behaviour: falls right hand
behaviour: falls

10. Odd degree polynomials fall on the left and rise on
the right hand sides of the graph (like x3) if the
coefficient is positive.
turning Points
in the middle right hand
behaviour: rises
left hand
behaviour: falls

11. Odd degree polynomials rise on the left and fall on
the right hand sides of the graph (like x3) if the
coefficient is negative.
turning points
in the middle
left hand
behaviour: rises
right hand
behaviour: falls