lecture # 3 mth 104 calculus and analytical geometry
TRANSCRIPT
Lecture # 3
MTH 104
Calculus and Analytical Geometry
Upward shift
Functions: Translation(i) Adding a positive constant c to a function y=f(x) ,adds c to each y-coordinate
of its graph, thereby shifting the graph of f up by c units.
Down ward shift
Functions: Translation
(ii) Subtracting a positive constant c from the function y=f(x) shifts the graph down by c units.
Functions: Translation
(iii) If a positive constant c is added to x , then the graph of f is shifted left by c units.
Left shift
Functions: Translation
(iii) If a positive constant c is subtracted from x , then the graph of f is shifted right by c units.
Right shift
Translations
Example Sketch the graph of
23 )( 3 )( xybxya
Translations
Translations
Translations
Translations
Translations
Reflection about y-axis
Functions: Reflection(i) The graph of y=f(-x) is the reflection of the graph of y=f(x) about the y-axis because the point (x,y) on the graph of f(x) is replaced by (-x,y).
Reflection about x-axis
Functions: Reflection(ii) The graph of y=-f(x) is the reflection of the graph of
y=f(x) about the x-axis because the point (x,y) on the graph of f(x) is replaced by (x,-y).
Functions: Stretches and Compressions
Multiplying f(x) by a positive constant c has the geometric effect of stretching the graph of f in the y-direction by a factor of c if c >1 and compressing it in the y-direction by a factor of 1/c if 0< c >1
Stretches vertically
Functions: Stretches and Compressions
Compresses vertically
Functions: Stretches and Compressions
Multiplying x by a positive constant c has the geometric effect of compressing the graph of f(x) by a factor of c in the x-direction if c > 1 and stretching it by a factor of 1/c if 0< c >1.
Horizontal compression
Functions: Stretches and Compressions
Horizontal stretch
Symmetry
Symmetry tests: • A plane curve is symmetric about the y-axis if and
only if replacing x by –x in its equation produces an equivalent equation.
• A plane curve is symmetric about the x-axis if and only if replacing y by –y in its equation produces
an equivalent equation.• A plane curve is symmetric about the origin if and
only if replacing both x by –x and y by –y in its equation produces an equivalent equation.
Symmetry
Example: Determine whether the graph has symmetric
about x-axis, the y-axis, or the origin.
5 )(
32 )( 95 )( 222
xyc
yxbyxa
Even and Odd function A function f is said to be an even function if
f(x)=f(-x)
And is said to be an odd function if f(-x)=-f(x)
Examples:
Even and Odd function
Polynomials
An expression of the form
is called polynomial, where a’s are constants and n is a non-negative integer. E.g.
axaxaxaxa nn
nn
nn
12
21
1
Rational functions
A function that can be expressed as a ratio of two polynomials is called a rational function. If P(x) and Q(x) are polynomials, then the domain of the rational function
Consists of all values of such that Q(x) not equal to zero.
Example:
)(
)()(
xQ
xPxf
Algebraic Functions
Functions that can be constructed from polynomials by applying finitely many algebraic operations( addition, subtraction, division, and root extraction) are called algebraic functions. Some examples are
Algebraic Functions
Classify each equation as a polynomial, rational, algebraic or not an algebraic functions.
22
2
3
4
43 )(
72
5 )(
4cos5 )(
13 )(
2 )(
xxye
x
xyd
xxyc
xxyb
xya
The families y=AsinBx and y=AcosBx
We consider the trigonometric functions of the form y=Asin(Bx-C) and y=Acos(Bx-C)
Where A, B and C are nonzero constants. The graphs of such functions can be obtained by stretching, compressing, translating, and reflecting the graphs of y=sinx and y=cosx. Let us consider the case where C=0, then we have
y=AsinBx and y=AcosBxConsider an equation y=2sin4x
The families y=AsinBx and y=AcosBx
Y=2sin4x
Amplitude=
Period=
The families y=AsinBx and y=AcosBx
In general if A and B are positive numbers, the graphs of y=AsinBx and y=AcosBx oscillates between –A and A and repeat every units that is amplitude is equal to A and period .
If A and B are negative, then
Amplitude= |A|, Period= frequency=
Example Find the amplitude, period and frequency of
B2
B2
B2
2B
xycxybxya sin1 )( )5.0cos(3 )( 2sin3 )(
The families y=Asin(Bx-C) and y=Acos(Bx-C)
These are more general families and can be rewritten as
y=Asin[B(x-C/B)] and y=Acos[B(x-C/B)]Example Find the amplitude and period of
)23
sin(4 )( )2
2cos(3 )( x
ybxya