1. Lecture # 3 MTH 104 Calculus and Analytical Geometry

2. 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.

3. Down ward shift Functions: Translation (ii) Subtracting a positive constant c from the function y=f(x) shifts the graph down by c units.

4. 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

5. 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

6. Translations Example Sketch the graph of

7. Translations

12. 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).

13. 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).

14. 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 1 Stretches vertically

15. Functions: Stretches and Compressions Compresses vertically

16. 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 1. Horizontal compression

17. Functions: Stretches and Compressions Horizontal stretch

18. 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.

19. Symmetry Example: Determine whether the graph has symmetric about x-axis, the y-axis, or the origin.

20. 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:

21. Even and Odd function

22. Polynomials An expression of the form is called polynomial, where a’s are constants and n is a non-negative integer. E.g.

23. 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:

24. 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

25. Algebraic Functions Classify each equation as a polynomial, rational, algebraic or not an algebraic functions.

26. 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=AcosBx Consider an equation y=2sin4x

27. The families y=AsinBx and y=AcosBx Y=2sin4x Amplitude= Period=

28. 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

29. 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