Name _________________________
Date _______________
7.7 – Higher Degree Polynomials
The shape of this graph is typical of the higherdegree polynomial graphs you will work with in
this lesson. Note that this graph has one local
maximum at (5, 2000) and one local minimum
at (15, 0). These are the points that are higher
or lower than all other points near them. You
can also describe the end behavior: what
happens to f(x) as x takes on extreme positive
and negative values. In the case of this cubic
function, as x increases, y also increases. As x
decreases, y also decreases.
Graphs of all polynomials with real coefficients
have one y-intercept. Polynomials with real
coefficients also possibly have one or more xintercepts, but may not have any. You can also
describe other features of polynomial graphs,
such as local maximums or minimums and end
behavior.
Example A: Find a polynomial function whose graph has x-intercepts 3, 5, and -4, and y-intercept
180. Describe the features of its graph and sketch it.
Example B: Can you think of another polynomial with the same x- and y-intercepts as Example A?
Describe the features of its graph and sketch it.

You can identify the degree of many polynomial functions by looking at the shapes of their graphs.
Every 3rd-degree polynomial function has essentially one of the shapes shown below. Graph A
shows the graph of y  x . It can be translated, stretched, or reflected. Graph B shows one
3
possible transformation of Graph A. Graphs C and D show the graphs of general cubic functions
3
2
in the form y  ax  bx  cx  d . In Graph C, a is positive, and in Graph D, a is negative.
You’ll explore the general shapes and characteristics of other higher-degree polynomials in the
investigation.
Mini Investigation:
The graph of y  2( x  3)( x  5)( x  4) has x-intercepts 3, 5, and -4 because they are the only
2
possible x-values that make y = 0. This is a 4th degree polynomial, but it has only three x-intercepts.
The root x = -4 is called a double root because the factor (x + 4) occurs twice.
a. Complete the missing table entries for polynomial functions ii-vi.
Polynomial function
Degree
i.
y  2( x  3)( x  5)( x  4)2
ii.
y  2( x  3)2 ( x  5)( x  4)
4
Roots
one at 3, one at 5, two at -4
iii. y  2( x  3)( x  5) ( x  4)
2
iv.
v.
two at 3, one at 5, two at -4
y  2( x  3)( x  5)( x  4)3
vi. y  2( x  3)( x  5) ( x  4)
2
3
b. Use your calculator to make a sketch of the general shape of each of the functions i-vi. Each
graph should display all relevant features, including local extreme values. (You will need to
change your window to see the entire graph shape.)
i. y  2( x  3)( x  5)( x  4)
2
iv.
ii. y  2( x  3) ( x  5)( x  4)
iii. y  2( x  3)( x  5) ( x  4)
v. y  2( x  3)( x  5)( x  4)
vi. y  2( x  3)( x  5) ( x  4)
2
3
2
2
3
_______________________________
c. Based on your graphs and the table, describe the connection between the power of a factor and
what happens at that x-intercept.
Single Roots
Double Roots
Triple Roots
Use what you learned about roots and sketch the general behavior of the following graphs.
1.
y  2( x  2)2 ( x  1)( x  5)3
Degree: _________
Single Root: __________
Double Root: _________
Triple Root: __________
End Behavior:
As x decreases, y _____________
As x increases, y ______________
2.
y  2( x  2)( x  1)2 ( x  4)2
Degree: _________
Single Root: __________
Double Root: _________
Triple Root: __________
End Behavior:
As x decreases, y _____________
As x increases, y ______________
3.
y  5( x  1)( x  5)( x  4)3
Degree: _________
Single Root: __________
Double Root: _________
Triple Root: __________
End Behavior:
As x decreases, y _____________
As x increases, y ______________