Liceo Scientifico “Isaac Newton”Roma

School Year 2011-2012

Maths course

Exponential function

XX

YY

OO

(0,1)

12M = M + M i = M ( 1 + i ) = C ( 1 + i )²

11

Sum of money

M = C + C i = C ( 1 + i )1

M = C ( 1 + i ) x

It’s great!

where the variable x can indicate also fractional values of time.

( 1 + i ) > 1

Compound interest

General exponential function

y = ax

a > 0

a > 1 0 < a < 1 a = 1

+Its domain is the set of real numbers R, while its codomain is the set of real positive numbers R

x є R

XX

YY

OO

(0,1)

00

x y

1

1

3

2

-2

-1-1

-3

4

2

1814

12

81

If a > 1

x y = 2

First condition

XX

YY

OO

(0,1)

1

y = 2x

y = 3x

Key:

00

x y

1

1

3

2

-2

-1-1

-3

4

2

18

14

12

8

y = 2- x

XX

YY

OO

(0,1)

1

if 0 < a < 1

Second condition

y = 2- x

y = 3- x

Key:

XX

YY

OO

(0,1)

1

If a = 1

XX

YY

OO

(0,1)

1

y = 1

Third condition

y = 2 x

y = 2- x

Key:

XX

YY

OO

(0,1)

1

The symmetry about the y axis

In fact if we apply the equations of symmetry about the y axis to the function y = a we obtain the following curve:

Y = a- x

y = a x X = - x

Y = y

This property is true for every pair of exponential functions of this type:

y = ax

y = a- x

x

the injectivity

the surjectivity

the exponential function is bijective

Properties of the exponential function

so it’s invertible

XX

YY

OO

(0,1)

(1,0)

If a > 1

y = log xa

y = a x

y = x

Logarithmic function

If 0 < a < 1

y = log xa

y = a x

XX

YY

(0,1)

(1,0)

y = x

OO

Logarithmic function

a > 1

Let C and C’ be two curves both passing through a point P; we say that C is steeper than C’ in P if the slope of the tangent in P to C is greater than the one of the tangent in P to C’.

DEFINITION

the exponential function grows faster than any polynomial function

XX

YY

OO

(0,1)

1

y = 2x

y = x + 2 x + 12

Key:

y = 2 x + 1

y = ln2 ∙ x + 1

P

Euler‘s number e

Values of aExponential

functionSlope m

a = 2 y = 2 m = 0.6931

a = 2.5 y = 2.5 m = 0.9163

a = 2.7 y = 2.7 m = 0.9933

a = 2.71 y = 2.71 m = 0.9969

a = 2.718 y = 2.718 m = 0.9999

a = 2.719 y = 2.719 m = 1.0003

x

x

x

x

x

x

this means that there’s a value of a between these two values for which the slope of the tangent in P is equal to 1.

a ≤ 2.718 then m < 1,

If a grows, m also grows

a ≥ 2.719 then m > 1;

The exponential function having base equals e is called a natural exponential function and its equation is:

y = ex

It is an irrational number and a transcendental number becauseit isn’t the solution of any polynomial equation with rational coefficients.

e = 2.7

This number is called Euler’s number e in honour of the mathematician who discovered it.

y = e x

y = x + 1

Key:

XX

YY

OO

(0,1)

1

y = ex

y = x + 1

45 °

Y = - a x

y = a x X = x

Y = - y

or compressions.2) translations, 3) dilations 1) symmetries,

Graphs of non-elementary exponential functions:

1) if we apply the equations of the symmetry about the x axis to the function y = a we obtain the following curve: x

2) if we apply the equations of a translation by a vector v having components (h,k) to the function y = a we obtain the following curve:

x

Y = a + kY – k = a X - h

y = a x X = x + h

Y = y + k

X - h

3) if we apply the equations of a dilation to the function y = a we obtain the following curve:

x

y = a x X = h x

Y = k y = a

Y

k

X

h

x

x1) the equation y = 2 + 1 represents the curve y = 2 shifted up by one:

y = 2x

y = 2 + 1 x

Key:

YY

(0,1)

1

y = 1

OO XX

y = 2x

y = 2 + 1x Examples of graphs:

y = 2x

y = 2x + 1

Key:

YY

(0,1)

1

x + 1

x2) the equation y = 2 represents the curve y = 2 shifted one point to the left:

XXOO

y = 2x

y = 2x + 1

y = -2

YY

(0,1)

x - 3

3) the equation y = 2 - 2 represents the curve y = 2 shifted three points to the right and shifted down by two:

x y = 2x

y = 2 - 2x - 3

Key:

XXOO 1

y = 2x

y = 2 - 2x - 3

4) the equation y = 3 ∙ 2 represents the curve y = 2 in which the abscissas don’t change while the ordinates are tripled:

x

x

y = 2x

y = 3 ∙ 2x

Key:

YY

(0,1)

1 XXOO

y = 2x

y = 3 ∙ 2x

XXOO

x5) the equation y = 2 represents the curve y = 2 in which the ordinates don’t change while the abscissas are tripled:

3

x

y = 2x

Key:

y = 2 3

x

YY

(0,1)

1

y = 2x

y = 2

x

3

x

y = 2x

y = 2- x

Key:

YY

(0,1)

1 XXOO

6) the equation y = 2 represents two curves:

y = 2 x

y = 2- x

if x ≥ 0

if x < 0y = 2

xy = 2

- x

XX

7) the equation y = 2 represents two curves:

x + 1

x + 1y = 2

y = 2- x - 1

if x ≥ -1

if x < -1

YY

(0,1)

OO 1-1

y = 2x +1

y = 2- x -1

Key:

XXOO-1

8) the equation y = 2 - 1 represents the graph of the previous curve number 7 shifted down by one:

x + 1

YY

(0,1)

1

y = 2 - 1 x +1

y = 2 - 1- x -1

Key:

9) the equation y = - 2 + 1 can be rewritten as follows:

x + 1

x + 1

this equation representsthe symmetrical curve of thefunction y = 2 - 1 aboutthe x axis

y = - ( 2 - 1 ) x + 1

YY

(0,1)

1OO

(0,-1)

XX

y = 2 - 1 x +1

Key:

y = - (2 - 1 )x +1

XX

10) the equation y = 2 - 2x + 1

YY

OO

(0,1)

1

y = 2 -2 x + 1

the part of the negative ordinates is substituted by its mirror image about the x axis:

.

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