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Advance Mathematics Objectives: Define Even and Odd functions algebraically and graphically Sketch graphs of functions using shifting, and reflection Section 3.5

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Even and Odd Functions TerminologyDefinitionExampleType of Symmetry f is an even function f(-x) = f(x)y = f(x) = x 2 w.r.t y-axis f is an odd function f(-x) = - f(x)y = f(x ) = x 3 w.r.t Origin Example 9 Determine whether a function is even, odd or neither. a) f ( x ) = 3x 4 + 5x 2 –4 b) f( x ) = -2x 5 +4x 3 +7x c) f( x ) = x3 x3 +x 2 Solution: a)Substitute x by –x f( -x) = 3( -x )4 )4 + 5 ( ) = 3x 4 + 5x 2 – 4 = f(x) f( -x) = f(x) f is even. Substitute x by –x f(-x) = -2 (-x) 5 + 4( -x )3 )3 +7(-x) = 2x 5 – 4x 3 – 7x = - (-2x 5 +4x 3 +7 ) = - f(x) f(-x) = - f(x) f is odd Substitute x by –x f(-x) = ( -x )5 )5 + ( )2)2 = - x5 x5 + x2x2 f(-x) is not equal to f( x) nor –f(x). Therefore, f is neither.

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Continue… Example 10 Check whether the following graphs represent an even or odd functions or neither. The graph represents an even functionThe graph represents neither The graph represents an odd function The graph represents neither a) b) c) d)

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Continue… Example 11 Complete the graph of the following if a) Symmetric w.r.t y-axis b) Symmetric w.r.t origin c) Function is even d) Function is odd e) Symmetric w.r.t x-axis Graph of (e ) does not represent a function

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Example 12 y = f( x ) + cUp c units y = f ( x ) - cDown c units Vertical Shifting Below is the graph of a function y = f ( x ). Sketch the graphs of a)y = f ( x ) + 1 b) y = f ( x ) - 2 y = f(x) + 1 y = f (x) y = f (x)-2

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Horizontal Shifting y = f( x + c )Left c units y = f ( x - c )Right c units Continued… Example 13. Given the graph of a function y = f ( x ). Sketch the graphs of a)y = f ( x + 3 ) b) y = f ( x – 4 ) y = f ( x ) y = f ( x ) + 3 y = f ( x ) - 4 Horizontal Shift 3 units to the left Horizontal Shift 4 units to the right

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Continued… Example 14 Can you tell the effects on the graph of y = f ( x ) y = f( x + h ) + k y = f( x + h ) - k y = f( x - h ) + k y = f( x - h ) - k Example 15 Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1 y = f( x ) y = f( x + 2 ) - 1 Left h units and Up k units Left h units and Down k units Right h units and Up k units Right h units and Down k units

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Continued … Example 16 Below is the graph of a function. Sketch the graph of y = f( x ) y =f(x-2)-3 The graph of the absolute value is shifted 2 units to the right and 3 units down Solution:

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Vertical Stretching y = cf( x) ( c> 1 )Vertical Stretch by a factor c y = (1/c)f ( x) ( c > 1 )Vertical Compress by a factor 1/c Note1 :When c > 1. Then 0 < 1/c < 1 Note 2 : c effects the value of y only. Example 17 Below is the graph of a function y = x2 x2. Sketch the graphs of 1.y = 5 x2x2 2. y = (1/5)x 2 xy = x 2 y=5x 2 y=1/5x

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Example 18 If the point P is on the graph of a function f. Find the corresponding point on the graph of the given function. 1)P ( 0, 5 ) y = f( x + 2 ) – 1 2)P ( 3, -1 ) y = 2f(x) +4 3) P( -2,4)y = (1/2) f( x-3) + 3 Solution: 1)P ( 0,5). y = f( x + 2 ) – 1 shifts x two units to the left and shifts y one unit down. The new x =0 – 2 = -2, and the new y = 5 – 1 = 4. The corresponding point is ( -2, 4 ). 2) P(3,-1). y = 2f(x) +4 has no effect on x. But it doubles the value of y and shifts it 4 units vertically up. Therefore the new x = 3(same as before ),), and the new value of y = 2 (-1 ) + 4 = 2. Therefore, the corresponding point is ( 3,2 ). 3)P(-2, 4 ). y = (1/2) f( x-3) + 3 shifts x 3 units to the right and splits the value of y in half and then shifts it 3 units up. That is, the new value of y = (1/2)(4) + 3 = 5. Therefore, the corresponding point is ( 1, 5 ).

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Reflecting a graph through the x-axis y = -f( x)Reflection through the x-axis (x-axis acts as a plane mirror) Example 19 Note: For any point P(x,y) on the graph of y = f(x), The graph of y = - f(x) does not effect the value of x, but changes the value of y into - y Below is the graph of a function y = x2 x2. Sketch the graph of 1.y = - x2x2 xy = x 2 y = -x

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Sketching a piece-wise function Example 20 Definition: Piece-wise function is a function that can be described in more than one expression. Sketch the graph of the function f if Solution: Graph y = 2x + 5 and take only the portion to the left of the line x = The point (-1, 3 ) is included. Graph y = x2 x2 and take only the portion where –1 < x < 1. Note: the points ( -1,1) and ( 1, 1 ) are not included Graph y = 2 and take only the portion to the right of x = 1. Note: y = 2 represents a horizontal line. The point (1, 2 ) is included.

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Sketching the graph of an equation containing an absolute Example 21 Sketch the graph of y = g ( x ) = Note: To sketch an absolute value function. We have to remember that And hence, the graph is always above the x-axis. The part of the graph that is below the x-axis will be reflected above the x-axis. Strategy: 1. Graph y = f(x) = x2.x2. Solution: 2. Graph y = f( x ) - 9 = x 2 – 9 by shifting the graph of f 9 units down 3. Graph g(x) by keeping the portion of the graph y = f( x ) - 9 = x 2 – 9 which is above the x-axis the same, and reflecting the portion where y < 0 with respect to the x-axis. 4. Delete the unwanted portion

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Example 22 Below is the graph of y = f(x). Graph Let the animation talk about itself Solution: A picture can replace 1000 words

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Do all homework exercises in the syllabus

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