Functions for Grade 10

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Functions for Grade 10

  1. 1. Functions Prepared by Boipelo Radebe Grade 10
  2. 2. Relation is referred to as any set of ordered pair. Conventionally, It is represented by the ordered pair ( x , y ). x is called the first element or x-coordinate while y is the second element or y-coordinate of the ordered pair. DEFINITIONDEFINITION
  3. 3. Relations are set of ordered pairs
  4. 4. Definition: Function •A function is a special relation such that every first element is paired to a unique second element. •It is a set of ordered pairs with no two pairs having the same first element.
  5. 5. Functions  Functions are relations, set of ordered pairs,in which the first elements are not repeated.
  6. 6. Function Notation •Letters like f , g , h and the likes are used to designate functions. •When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f . •The notation f ( x ) is read as “ f of x ”.
  7. 7. Graph of a Function •If f(x) is a function, then its graph is the set of all points (x,y) in the two-dimensional plane for which (x,y) is an ordered pair in f(x) •One way to graph a function is by point plotting. •We can also find the domain and range from the graph of a function.
  8. 8. DEFINITION: Domain and RangeDEFINITION: Domain and Range • All the possible values of x is called the domain. • All the possible values of y is called the range. • In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively.
  9. 9. Domain and range of a function
  10. 10. 7 Function Families  What you need to know:  Name  Equation  Domain  Range
  11. 11. Linear  Name – Constant  Equation –  Domain – (-∝,∝)  Range – [b] y b=
  12. 12. Linear  Name – Oblique Linear  Equation –  Domain – (-∝,∝)  Range – (-∝,∝) y m x b= +
  13. 13. Power Functions  Name – Quadratic  Equation –  Domain – (-∝,∝)  Range – [0,∝) y x= 2
  14. 14. Reciprocal Functions  Name – Rational  Equation –  Domain –(-∝,0)∪(0,∝)  Range – (-∝,0) ∪ (0,∝) y x = 1
  15. 15. Power functions  Name - exponential  Equation – y= a  Domain – (-∝,∝)  Range – (0, ∝) x
  16. 16. Vertical Line Test  A curve in the coordinate plane is the graph of a function if no vertical line intersects the curve more than once.
  17. 17. Graphs of functions?
  18. 18. Increasing and Decreasing Functions A function f is increasing if:  A function f is decreasing if: f x f x w h e n x x ( ) ( )1 2 1 2 < < f x f x w h e n x x ( ) ( )1 2 1 2 > <
  19. 19. State the intervals on which the function whose graph is shown is increasing or decreasing.
  20. 20. Transformations Vertical Shift Horizontal Shift Reflecting Stretching/Shrinking
  21. 21. General Rules for Transformations  Vertical shift:  y=f(x) + c ⇒ c units up  y=f(x) – c ⇒ c units down  Horizontal shift:  y=f(x+c) ⇒ c units left  y=f(x-c) ⇒ c units right  Reflection:  y= – f(x) ⇒ reflect over x-axis  y= f(-x) ⇒ reflect over y-axis  Stretch/Shrink:  y=af(x) ⇒ (a > 1) Stretch vertically  y=af(x) ⇒ (0 < a < 1) Shrink vertically
  22. 22. Exploring transformations  Graph o Graph o Graph o Graph y x= 2 y x y x y x y x y x y x = + = − = − = + = = 2 2 2 2 2 2 3 2 4 3 2 1 2 ( ) ( ) { { {
  23. 23. Even & Odd Functions  Algebraically:  Even – f is even if f(-x) = f(x)  Odd – f is odd if f(-x) = - f(x)  Graphically:  Even – f is even if its graph is symmetric to the y- axis  Odd – f is odd if its graph is symmetric to the origin
  24. 24. Use the rules of transformations to graph the following: ( ) y x y x y x y x y x = − + = + − = − − = − + − = − + 2 3 2 1 2 4 3 2 6 1 3 1 2 5 2 3 ( )
  25. 25. Trigonometric Functions Name – Sine Equation -y = a sin bx + c Domain - (-∝,∝) Range – [ 1. -1 ] amplitude = a period = b 360° phase shift = b Vertical shift =c
  26. 26. Trigonometric Functions Name – Cosine Equation - y = acosbx + c amplitude = a period = b 360° phase shift = b Vertical shift =c Domain - (-∝,∝) Range – [ 1. -1 ]
  27. 27. Trigonometric Functions Name – tangent (tan) Equation -y = a tan bx + c amplitude = a period = b 180° phase shift = b Vertical shift =c Domain – x = - 180, -90, 90, 180 Range – (-∝,∝)
  28. 28. Graphs of functions in real life
  29. 29. Parabolas in life
  30. 30. Parabolic building
  31. 31. Do the following work on your own.
  32. 32. EXAMPLE 1 Evaluate each function value 1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ? 2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )? 3. If h ( x ) = x 2 + 5 , find h ( x + 1 ). 4.If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 , Find: a) f(g(x)) b) g(f(x))
  33. 33. Example 2 Graph each of the following functions. 5x3y.1 −= 1.2 += xy 2 x16y.3 −= 5xy.4 2 −= 3 x2y.5 = x 5x3 y + = 4xy.7 −−= 6.
  34. 34. Example 3 Determine Algebraically if the function is even, odd or neither y x x y x x y x x y x x x = + = − = − = − + + 2 6 2 3 3 2 4 3 5 2 4 3 1
  35. 35. Reference  Gurl, V . 2010. Afm chapter 4. functions. http://www.slideshare.net/volleygurl22/afm-chapter-4-powerpoint?qid=e6c from_search=1. Accessed 06 March 2014  Manarang, K . 2011. 7 Functions. http://www.slideshare.net/KathManarang/7-functions- 9175161. Accessed on 06 March 2014  Farhana S .2013. Graphs and their functions. http://www.slideshare.net/farhanashaheen1/function- and-their-graphs-ppt?qid=e22cda30-fde3-4c4a-b233- f00ff6f20596&v=default&b=&from_search=2. Accessed on 06 March 2014  Schmitz, T .2008.Higher Maths 1.2.3 - Trigonometric Functions. http://www.slideshare.net/timschmitz/higher- maths-123-trigonometric-functions-358346?qid=4e5bcb29- 5942-48aa-9735-bf4c30ac5f05&v=qf1&b=&from_search=1. Accessed on 06 March 2014
  36. 36. Thank you

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