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Basic Functions

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Basic Functions

Polynomials
Exponential Functions
Trigonometric Functions

lim

x →0

sin ( x )
x

=1

Trigonometric Identities
The Number e

Index

FAQ

Polynomials
Definition

Polynomial is an expression of the type
P = a0 + a1x + a2 x 2 + K + an x n
where the coefficients a0 , a1,K, an are real numbers and an ≠ 0.

The polynomial P is of degree n.
A number x for which P(x)=0 is called a root of the
polynomial P.

Theorem

Index

A polynomial of degree n has at most n real roots.
Polynomials may have no real roots, but a polynomial of
an odd degree has always at least one real root.

Mika Seppälä: Basic
Functions

FAQ

Graphs of Linear Polynomials
Graphs of linear polynomials y = ax + b are straight lines. The coefficient
“a” determines the angle at which the line intersects the x –axis.

Graphs of the linear
polynomials:
1. y = 2x+1 (the red line)
2. y = -3x+2 (the black line)
3. y = -3x + 3 (the blue line)

Index

Mika Seppälä: Basic
Functions

FAQ

Graphs of Higher Degree
The behaviour of a polynomial
Polynomials P = a + a x + K + a x
0

1

n

n

for large positive

or negative values x is determined by the highest degree term "an x n ".

If an > 0 and n is odd, then as x → ∞ also P ( x ) → ∞.
Likewise: as x → −∞ also P ( x ) → −∞.

If an > 0 and n is even, then as x → ±∞, P ( x ) → ∞.
Problem

The picture on the right shows the
graphs and all roots of a 4th degree
polynomial and of a 5th degree
polynomial. Which is which?

Solution

The blue curve must be the graph of
the 4th degree polynomial because
of its behavior as x grows or gets
smaller.

Index

Mika Seppälä: Basic
Functions

FAQ

Measuring of Angles (1)
Angles are formed by two half-lines starting
from a common vertex. One of the half-lines
is the starting side of the angle, the other one
is the ending side. In this picture the starting
side of the angle is blue, and the red line is
the ending side.
Angles are measured by drawing a circle
of radius 1 and with center at the vertex
of the angle. The size, in radians, of the
angle in question is the length of the
black arc of this circle as indicated in the
picture.
In the above we have assumed that the angle is
oriented in such a way that when walking along
the black arc from the starting side to the
ending side, then the vertex is on our left.
Index

Mika Seppälä: Basic
Functions

FAQ

Measuring of Angles (2)
The first picture on the right shows a
positive angle.

α

The angle becomes negative if the orientation
gets reversed. This is il...
Index
FAQ
Basic Functions
Polynomials
Exponential Functions
Trigonometric Functions
Trigonometric Identities
The Number e
( )
0
sin
lim 1
x
x
x
=
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