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Chapter 5: Section 5.1
Polynomials
Section 5.1: Polynomials
Definition: A Function is a rule that assigns to each input value x exactly out-
put value y = f(x). The variable x is called the variable, and y is called the
variable. The is the set of all allowable x values, and the
is the set of all possible y values.
When graphing a function, the horizontal (x) axis is the domain and the vertical (y) axis is the range.
Example 1, Reading the information from a graph: The graph of a function y = f(x) is shown
in the following figure
(a) Find the values of f(�2) and f(0).
(b) Determine the value of x that makes f(x) = 1
�3 �2 �1 0 1 2 3
�3
�2
�1
1
2
3
x
f(x)
Vertical Line Test: A curve in the xy-plane is the graph of a i↵ no vertical line
passes through the curve more than .
independentdependent domain range
fl . 2) = - I
f ( o )= I
f( ⇒ . ,
TWO X Values
I g
that make f(×)=1
q° . xio six =3→
functiononce
Chapter 5: Sec5.1, Polynomials
Example 2: Determine whether each is a function of x.
(a)
x
f(x)
(b)
�4�3�2�1 0 1 2 3 4�1
1
2
3
4
x
f(x)
(c)
x
f(x)
2 Spring 2018, Maya Johnson
|||||||1 s a function
|/||||/' saturation
HH Fitton
Chapter 5: Sec5.1, Polynomials
In order to describe the domain and range of a function, we often use interval notation.
means all numbers between a and b, not including the endpoints
means all numbers between a and b, including the endpoints
Example 3: Determine the domain and range of the following function.
�3 �2 �1 0 1 2 3
�3
�2
�1
1
2
3
x
f(x)
Example 4: Let f(x) = x2+ 2x� 1, find the following
(a) f(1)
(b) f(x+ h)
(c) f(x+ h)� f(x)
3 Spring 2018, Maya Johnson
( a, b)
{ a ,b ]
-- Domain : too
,3 ]
Range
:(- a
,3 ]
; .
*- oo
. x
= ( 1) 2+2 C 1) - 1
= I + 2 -I = �2�
=
lxth5t2lxthDa1olXthXxthDt2xt2h-1x2t2xhth2t2xt2TixYt2xhth2tYxt2hf-CxXtI-2xhth2t@A1wayson1gtecmswith.h
"
leftover
Chapter 5: Sec5.1, Polynomials
(d)f(x+ h)� f(x)
h...The Di↵erence Quotient
A function of the form f(x) = ax2+ bx+ c
is called a function, where a, b, and c are real numbers and a 6= 0. This is the
standard form.
The graph of a quadratic function is called a
The highest (Maximum) or lowest (Minimum) point on a parabola is called the
of the quadratic function. It is the point (h, k) where h =�b
2aand k = f(h) (the quadratic function
evaluated at x = h)
Example 5: Given y = f(x) = �4x2+ 16x� 8,
(a) Find the vertex
4 Spring 2018, Maya Johnson
=
'
- ÷a= ¥45 # = 2
K = fl 2) = -
4125+1614-8=8
Vertex =l2,@
Chapter 5: Sec5.1, Polynomials
(b) Find the maximum value of the function
(c) Find the minimum value of the function
(d) What are the domain and range of the function?
The real (or roots) of a function are its x-intercepts. There are several ways to find
the zeros of a quadratic function, but if you are not a fan of factoring and the quadratic function is in
standard form, then you can use the quadratic formula:
x =�b±
pb2 � 4ac
2a
Example 6: Use the quadratic function y = 2x2+ 2x� 24 to answer the following.
(a) Find the vertex. Is the y-coordinate of the vertex a maximum or a minimum value of the quadratic
function?
5 Spring 2018, Maya Johnson
a ' - 4 < 0 ⇒ vertex is a max
So Max value is �8�
No min vahue
÷theocracieszeros
.
Vertex :
h=;b== It , ¥ =
- I12
= ft - 42 ) =- 24.5
V{rtexist4z,-24==2 > 0
,so y
- cooed . of vertex
is a mdn not a max .
Chapter 5: Sec5.1, Polynomials
(b) Find the zeros of the function
Applications of Quadratic Functions: Quadratic Revenue and Profit Functions
We have previously defined the revenue function to be R(x) = sx (revenue equals selling price times
number of units sold). We also discussed that we can use a linear demand function to model the re-
lationship between quantity demanded and unit price. Combining those two concepts gives a revenue
function of the form:
R(x) = xp(x)
If you multiply a linear function by x, you get a quadratic function.
Example 7: It is found that the consumers of a particular toaster will demand 64 toaster ovens when
the unit price is $35 whereas they will demand 448 toaster ovens when the unit price is $5. Assuming
that the demand function is linear and the selling price is determined by the demand function,
a) Find the demand equation.
b) Find the revenue function.
c) Find the number of items sold that will give the maximum revenue. What is the maximum
revenue?
6 Spring 2018, Maya Johnson
× = -2t.az#2IIssg@. 2)
=
-2+-71 ⇒ x.tt# or -2yd
( 64135 ),
( 448,5 )35=624 ( 64 ) t b
slope = 35-5 3.55
¥,
+ b ⇒ b= 40If
= ¥ 11
[email protected]#=-yIyx2t@;teIitndmItFzknnYzda=gIe.=zs6toastJ
Max Rev .
=K=R(256
:)=$¥z¥
Chapter 5: Sec5.1, Polynomials
d) Find the unit price when the revenue is maximized.
e) If the company has a fixed cost of $1,000 and a variable cost of $15 per toaster, find the company’s
linear cost function.
f) What is the company’s maximum profit?
g) How many toasters should be sold for the company to break even?
7 Spring 2018, Maya Johnson
p ( 256 ) =D ( 256 ) =
$20C = ex + F =15x+@
P ( × )= Rlx )
- C ( x ) = jsqxr +
40×-15×-1000= gIyx2
+25×-1000
For max,
find vertex : ( h, k )
h=z±a=I¥%aJ
a 160 items
Max profit =k=P( 160 ) =$1@P ( x ) = 0
.
Find zeros of- Fy x 2+25×-1000
× = .tn#yFETooT2 (-5/64)
* = - 25 t.RS
Tea L3atgadIE←⇒tobreaken=€Round to nearest whole
numbers
Chapter 5: Sec5.1, Polynomials
A function of degree n is a function of the form:
f(x) = an xn+ an�1 x
n�1+ an�2 x
n�2+ . . .+ a1 x+ a0.
where an 6= 0 and the coe�cients, a0, a1, a2, . . . , an are real numbers. The number n is a nonnegative
integer and the coe�cient an is known as the leading coe�cient.
In general, there is not formula for finding the zeros of a polynomial function. However, there are
special cases when we can still find the zeros by hand.
Example 8: Find the zeros of the following polynomials:
(a) f(x) = x3 � 2x3 � 8x
(b) g(x) = (x2+ x� 2)(x2 � 9)
8 Spring 2018, Maya Johnson
polynomial
factor
x ( x2 - 2×-8 ) = O
X =O or XZ 2×-8=0
Solve × ? 2×-8=0
Can use quadratic formula) but I work factor
( x - 4) ( × + 2) =o
⇒ × - 4=0 or × +2=0
⇒×=4or×=-2or#lx2+x. 2) ( x2 . 9) =0
X2tx - 2=0 or × 2-9=0
solve ×2 + X - 2=0 ← factor ig( x + 2) ( × - 1) =o
⇒ X = - 2 or I
solve × 2-9=0+9 +9
× 2=9
⇒ x=±rg
⇒ x=±3
X=-2,1,-3=_
