functions for grade 10
TRANSCRIPT
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Functions
Prepared by Boipelo Radebe
Grade 10
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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
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Relations are set of ordered pairs
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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.
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Functions
Functions are relations, set of ordered pairs,in which the first elements are not repeated.
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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 ”.
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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.
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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.
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Domain and range of a function
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7 Function Families
What you need to know: Name
Equation
Domain
Range
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Linear
Name – Constant
Equation –
Domain – (-,)
Range – [b]
y b
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Linear
Name – Oblique Linear
Equation –
Domain – (-,)
Range – (-,)
y m x b
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Power Functions
Name – Quadratic
Equation –
Domain – (-,)
Range – [0,)
y x 2
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Reciprocal Functions
Name – Rational
Equation –
Domain –(-,0)(0,)
Range – (-,0) (0,)
yx
1
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Power functions
Name - exponential
Equation – y= a
Domain – (-,)
Range – (0, )
x
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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.
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Graphs of functions?
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Increasing and Decreasing Functions
A function f is increasing if:
A function f is decreasing if:
f x f x w hen
x x
( ) ( )1 2
1 2
f x f x w hen
x x
( ) ( )1 2
1 2
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State the intervals on which the function whose graph is shown is increasing or decreasing.
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Transformations
Vertical ShiftHorizontal ShiftReflectingStretching/Shrinking
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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
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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
( )
( )
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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
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Use the rules of transformations to graph the following:
y x
y x
y x
y x
yx
2 3 2
1
24 3
2 6
1 3
1
25
2
3
( )
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Trigonometric Functions
Name – Sine
Equation -y = a sin bx + c
Domain - (-,)
Range – [ 1. -1 ]
amplitude = a
period =b
360°
phase shift = bVertical shift
=c
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Trigonometric Functions
Name – Cosine
Equation - y = a cos bx + c
amplitude = a
period =b
360°
phase shift = bVertical shift
=c
Domain - (-,)
Range – [ 1. -1 ]
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Trigonometric Functions
Name – tangent (tan)
Equation -y = a tan bx + c
amplitude = a
period =b
180°
phase shift = bVertical shift
=c
Domain – x = - 180, -90, 90, 180
Range – (-,)
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Graphs of functions in real life
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Parabolas in life
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Parabolic building
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Do the following work on your own.
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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))
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Example 2Graph each of the following functions.
5x3y.1
1.2 xy
2x16y.3
5xy.4 2
3x2y.5
x
5x3y
4xy.7
6.
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Example 3Determine 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
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Reference Gurl, V . 2010. Afm chapter 4. functions.
http://www.slideshare.net/volleygurl22/afm-chapter-4-powerpoint?qid=e6cd91f5-5e87-4fa0-be23-f1afeb86873d&v=default&b=&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
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Thank you