lecture # 3 mth 104 calculus and analytical geometry
TRANSCRIPT
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Lecture # 3
MTH 104
Calculus and Analytical Geometry
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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.
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Down ward shift
Functions: Translation
(ii) Subtracting a positive constant c from the function y=f(x) shifts the graph down by c units.
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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
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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
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Translations
Example Sketch the graph of
23 )( 3 )( xybxya
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Translations
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Translations
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Translations
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Translations
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Translations
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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).
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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).
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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
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Functions: Stretches and Compressions
Compresses vertically
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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
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Functions: Stretches and Compressions
Horizontal stretch
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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.
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Symmetry
Example: Determine whether the graph has symmetric
about x-axis, the y-axis, or the origin.
5 )(
32 )( 95 )( 222
xyc
yxbyxa
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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:
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Even and Odd function
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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
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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
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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
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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
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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
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The families y=AsinBx and y=AcosBx
Y=2sin4x
Amplitude=
Period=
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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 )(
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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