Graphing polynomials

1. 7.2 Graphing Polynomial
Functions

2. To graph the function, make a table
of values and plot the corresponding
points. Connect the points with a
smooth curve and check the end
behavior.

3. GRAPHING POLYNOMIAL FUNCTIONS
END BEHAVIOR FOR POLYNOMIAL FUNCTIONS
CONCEPT
SUMMARY
> 0evenf (x)+ ¥ f (x) + ¥
> 0oddf (x)– ¥ f (x) + ¥
< 0evenf (x)– ¥ f (x) – ¥
< 0oddf (x)+ ¥ f (x) – ¥
a
n
n x – ¥ x +¥

4. x
f(x)
–3
–7
–2
3
–1
3
0
–1
1
–3
2
3
3
23
Graphing Polynomial Functions
Graph f (x) = x
3
+ x
2
– 4 x – 1.
SOLUTION
To graph the function, make a table of
values and plot the corresponding points.
Connect the points with a smooth curve
and check the end behavior.
The degree is odd and the leading coefficient is positive,
so f (x)–as x–and f (x)+as x+.

5. x
f (x)
–3
–21
–2
0
–1
–1
0
0
1
3
2
–16
3
–105
The degree is even and the leading coefficient is negative,
so f (x)–as x–and f (x)–as x+.
Graphing Polynomial Functions
Graph f (x) = –x
4
– 2x
3
+ 2x
2
+ 4x.
SOLUTION
To graph the function, make a table of
values and plot the corresponding points.
Connect the points with a smooth curve
and check the end behavior.

6. Extrema
•Turning points – where the graph of a function changes
from increasing to decreasing or vice versa
•Local maximum point – highest point or “peak” in an
interval
–function values at these points are called local maxima
•Local minimum point – lowest point or “valley” in an
interval
–function values at these points are called local minima
•Extrema – plural of extremum, includes all local maxima
and local minima

7. Local Maximum and Minimum
•f(a) is a local maximum (plural, local
maxima) if there is an interval around a
such that f(a) > f(x) for all values of x in the
interval, where x does not equal a.
•f(a) is a local minimum (plural, local
minima) if there is an interval around a such
that f(a) < f(x) for all values of x in the
interval, where x does not equal a.

8. Local Maximum and Minimum
In other words…
Local maxima are peaks
Local minima are valleys

9. For the following polynomial function,
find any local maxima or minima. Find the intervals
over which the function is increasing and decreasing.