Section 2.1
Quadratic Functions
and Polynomials
EXAMPLE
Suppose a baseball is dropped
from the top of the Empire
State Building. The table to the
right shows (based on
measurements) the distance d
(in feet) the baseball has fallen
after t seconds.
t (sec)
d (feet)
0
0
1
16
2
64
3
144
4
256
5
400
A FORMULA FOR
DROPPED BASEBALL
Note that d is not a linear
functions of t.
The distance d is given by the
formula
d=
16t2.
t (sec)
d (feet)
0
0
1
16
2
64
3
144
4
256
5
400
QUADRATIC FUNCTIONS
Definition: A quadratic function is one of the
form
f (x) = ax2 + bx + c
with a ≠ 0, so its formula involves a square term
as well as linear and constant terms.
GRAPHS OF QUADRATIC
FUNCTONS
Graph the following quadratic functions on your
calculator.
f (x) = x2
f (x) = -x2
f (x) = 3x2
f (x) = -2x2
f (x) = 6x2
f (x) = -0.75x2
f (x) = 0.5x2
f (x) = x2 – x – 6
f (x) = 0.1x2
f (x) = – 2x2 + 3x + 1
PARABOLAS
The graph of a quadratic function is a parabola.
The low point (or high point) is called the
vertex. The vertex is also the place where the
graph changes direction.
The quadratic function
f (x) = ax2 + bx + c
b
has its vertex at the x-value x  
.
2a
To find the y-coordinate, substitute this value in
for x.
POWER FUNCTIONS
Definition: A power function is one of the
form
f (x) = xk
where k is a given constant, which can either
be positive or negative, either an integer or a
fraction.
GRAPHS OF POWER
FUNCTIONS
Graph the following power functions.
f (x) = x2
f (x) = x
f (x) = x1/2
f (x) = x4
f (x) = x3
f (x) = x1/3
f (x) = x6
f (x) = x5
POLYNOMIALS
Definition: A polynomial function (or
simply a polynomial) is a sum of constant
multiples of the power functions
1, x, x2, x3, . . . xn, . . .
The degree of the polynomial is equals the
exponent of the highest power appearing
among its terms.
THE “END BEHAVIOR”
OF A POLYNOMIAL
The end behavior of a graph is what the shape
of the graph is on the extreme left and
extreme right.
The end behavior of a polynomial is the same
the end behavior of the term with the highest
power.
TURNING POINTS
OF POLYNOMIALS
In general, the graph of a polynomial can have as
many
degree - 1
turning points (or “bends”).
In other words, if the degree of the polynomial is
n, the graph can have as many as n - 1 turning
points (or “bends”).
SOLVING POLYNOMIAL
EQUATIONS
Principle: Graphic Solution of Equations
The solutions of the equation
f (x) = 0
are precisely the x-intercepts of the graph
y = f (x).
Recall, x-intercepts are the places where the
graph crosses (or touches) the x-axis.