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## Functions for Grade 10Presentation Transcript

• 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
• Relations are set of ordered pairs View slide
• 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. View slide
• Functions  Functions are relations, set of ordered pairs,in which the first elements are not repeated.
• 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 ”.
• 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.
• 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.
• Domain and range of a function
• 7 Function Families  What you need to know:  Name  Equation  Domain  Range
• Linear  Name – Constant  Equation –  Domain – (-∝,∝)  Range – [b] y b=
• Linear  Name – Oblique Linear  Equation –  Domain – (-∝,∝)  Range – (-∝,∝) y m x b= +
• Power Functions  Name – Quadratic  Equation –  Domain – (-∝,∝)  Range – [0,∝) y x= 2
• Reciprocal Functions  Name – Rational  Equation –  Domain –(-∝,0)∪(0,∝)  Range – (-∝,0) ∪ (0,∝) y x = 1
• Power functions  Name - exponential  Equation – y= a  Domain – (-∝,∝)  Range – (0, ∝) x
• 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.
• Graphs of functions?
• 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 > <
• State the intervals on which the function whose graph is shown is increasing or decreasing.
• Transformations Vertical Shift Horizontal Shift Reflecting Stretching/Shrinking
• 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
• 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 ( ) ( ) { { {
• 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
• 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 ( )
• 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
• Trigonometric Functions Name – Cosine Equation - y = acosbx + c amplitude = a period = b 360° phase shift = b Vertical shift =c Domain - (-∝,∝) Range – [ 1. -1 ]
• 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 – (-∝,∝)
• Graphs of functions in real life
• Parabolas in life
• Parabolic building
• Do the following work on your own.
• 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))
• 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.
• 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
• 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
• Thank you