* Identify power functions. * Identify end behavior of power functions. * Identify polynomial functions. * Identify the degree and leading coefficient of polynomial functions.

1. 5.2 Power Functions &
Polynomial Functions
Chapter 5 Polynomial and Rational Functions

2. Concepts and Objectives
⚫ Objectives for the section are
⚫ Identify power functions.
⚫ Identify end behavior of power functions.
⚫ Identify polynomial functions.
⚫ Identify the degree and leading coefficient of
polynomial functions.

3. Power Functions
⚫ A power function is a function with a single term that is
the product of a real number (the coefficient) and a
variable raised to a fixed real number.
⚫ It can be represented in the form
where k and p are real numbers.
( )= ,
p
f x kx

4. Power Functions (cont.)
⚫ Examples of power functions:
⚫
⚫
⚫
⚫ Non-examples of power functions:
⚫
⚫
( ) 3
f x x
=
( ) 2
f x x
= ( )
1
2
x
( ) 2
1
f x
x
= ( )
2
x−
( ) 2
4 1
f x x
= −
( )
5
2
2 1
3 4
x
f x
x
−
=
+

5. Polynomial Functions
⚫ A polynomial function consists of either zero or the sum
of a finite number of non-zero terms, each of which is a
product of a number (the coefficient) and a variable
raised to a non-negative integer (≥1) power
⚫ Let n be a non-negative integer. A polynomial function is
a function that can be written in the form
⚫ This is called the general form of a polynomial function.
Each ai is a coefficient and can be any real number, but
an ≠ 0.
( ) 2
2 1 0
n
n
f x a x a x a x a
= + + + +

6. Polynomial Functions (cont.)
⚫ Polynomial functions are different from power functions
in that
⚫ they can consist of multiple terms combined by
addition or subtraction
⚫ their exponents must be non-negative integers

7. The Degree and Leading Term
⚫ The degree of a polynomial (or power) function is the
highest power of the variable that occurs in the
polynomial.
⚫ The leading term is the term containing the highest
power of the variable.
⚫ The leading coefficient is the coefficient of the leading
term.
⚫ The leading term does not have to be the first term in
the polynomial.

8. Identifying the Degree and Leading
Coefficient
Examples: Identify the degree and the leading coefficient
of the following polynomial functions.
⚫ f
⚫
⚫
( ) 5 3
6 2 7
f t t t t
= − +
( ) 2 3
3 2 4
g x x x
= + −
( ) 3
6 2
h p p p
= − −

9. Identifying the Degree and Leading
Coefficient
Examples: Identify the degree and the leading coefficient
of the following polynomial functions.
⚫ f
Degree: 5 Leading coefficient: 6
⚫
Degree: 3 Leading coefficient: ‒4
⚫
Degree: 3 Leading coefficient: ‒1
( ) 5 3
6 2 7
f t t t t
= − +
( ) 2 3
3 2 4
g x x x
= + −
( ) 3
6 2
h p p p
= − −

10. Graphs of Polynomial Functions
⚫ If we look at graphs of functions of the form ,
we can see a definite pattern:
( )= n
f x ax
( )= 2
f x x ( )= 3
g x x
( )= 4
h x x ( )= 5
j x x

11. End Behavior
⚫ The end behavior of a polynomial graph is determined by
the leading term (also called the dominating term).
⚫ For example, has the same end
behavior as once you zoom out far enough.
( )= − +
3
2 8 9
f x x x
( )= 3
2
f x x

12. End Behavior
⚫ Example: Use symbols for end behavior to describe the
end behavior of the graph of each function.
1.
2.
3.
( )= − + + −
4 2
2 8
f x x x x
( )= + − +
3 2
2 3 5
g x x x x
( )= − + +
5 3
2 1
h x x x

13. End Behavior
⚫ Example: Use symbols for end behavior to describe the
end behavior of the graph of each function.
1.
2.
3.
( )= − + + −
4 2
2 8
f x x x x even function
opens downward
( )= + − +
3 2
2 3 5
g x x x x odd function
increases
( )= − + +
5 3
2 1
h x x x odd function
decreases

14. Turning Points and Intercepts
⚫ The point where a graph changes direction (“bounces”
or “wiggles”) is called a turning point of the function.
⚫ The y-intercept is the point at which the function has an
input value of zero.
⚫ The x-intercepts are the points at which the output value
is zero.

15. Determining Intercepts
⚫ To find the y-intercept, substitute 0 for every x.
⚫ To find the x-intercept(s), set the polynomial equal to
zero and solve for x.
Example: Find the intercepts of
⚫ y-intercept:
⚫ x-intercepts:
( ) ( )( )( )
2 1 4
f x x x x
= − + −
( ) ( )( )( )
( )( )( )
0 0 2 0 1 0 4
2 1 4 8
f = − + −
= − − =
( )( )( )
0 2 1 4
x x x
= − + −
2 0
2
x
x
− =
=
or
1 0
1
x
x
+ =
= −
or
4 0
4
x
x
− =
=

16. Determining Intercepts (cont.)
⚫ Desmos can be very handy for this as well. Compare the
intercepts we found with the graph of the function:

17. Turning Points and Intercepts
⚫ A polynomial function of degree n will have at most
n x-intercepts and n – 1 turning points, with at least one
turning point between each pair of adjacent zeros.
(Because it is a function, there is only one y-intercept.)
⚫ Example: Without graphing the function, determine the
local behavior of the function by finding the maximum
number of x-intercepts and turning points for
⚫ The polynomial has a degree of 10, so there are at
most 10 x-intercepts and at most 9 turning points.
( ) 10 7 4 3
3 4 2
f x x x x x
= − + − +

18. Turning Points and Intercepts
⚫ What can we conclude about the polynomial
represented by this graph based on its intercepts and
turning points?

19. Turning Points and Intercepts
⚫ What can we conclude about the polynomial
represented by this graph based on its intercepts and
turning points?
• The end behavior of the graph tells
us this is the graph of an even-
degree polynomial with a positive
leading term.
• The graph has 2 x-intercepts,
suggesting a degree of 2 or greater,
and 3 turning points, suggesting a
degree of 4 or greater. Based on
this, it is reasonable to conclude that
the degree is even and at least 4.

20. Putting It All Together
⚫ Use the information below about the graph of a
polynomial function to determine the function. Assume
the leading coefficient is 1 or ‒1.
The y-intercept is (0, ‒4). The x-intercepts are (‒2, 0),
(2, 0). Degree is 2. End behavior:

21. Putting It All Together
⚫ Use the information below about the graph of a
polynomial function to determine the function. Assume
the leading coefficient is 1 or ‒1.
The y-intercept is (0, ‒4). The x-intercepts are (‒2, 0),
(2, 0). Degree is 2. End behavior:
( ) ( )( )
2 2
f x x x
= + −

22. Classwork
⚫ College Algebra 2e
⚫ 5.2: 12-24 (even); 5.1: 26-32, 40-44 (even); 4.3: 32-
42 (even)
⚫ 5.2 Classwork Check
⚫ Quiz 5.1