Lecture # 3 MTH 104 Calculus and Analytical Geometry.

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