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**Advance Mathematics Section 3.5 Objectives:**

Define Even and Odd functions algebraically and graphically Sketch graphs of functions using shifting, and reflection

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**f(-x) = -2 (-x)5 + 4( -x )3 +7(-x) f(-x) = ( -x )5 + ( -x )2**

Even and Odd Functions Terminology Definition Example Type of Symmetry f is an even function f(-x) = f(x) y = f(x) = x2 w.r.t y-axis odd function f(-x) = - f(x) y = f(x ) = x3 Origin Example 9 Determine whether a function is even, odd or neither. a) f ( x ) = 3x4 + 5x2 – b) f( x ) = -2x5 +4x3 +7x c) f( x ) = x3 +x2 Substitute x by –x f(-x) = -2 (-x)5 + 4( -x )3 +7(-x) = 2x5 – 4x3 – 7x = - (-2x5 +4x3 +7 ) = - f(x) f(-x) = - f(x) f is odd Solution: Substitute x by –x f(-x) = ( -x )5 + ( -x )2 = - x5 + x2 f(-x) is not equal to f( x) nor –f(x). Therefore, f is neither. Substitute x by –x f( -x) = 3( -x )4 + 5 ( -x )2 - 4 = 3x4 + 5x2 – 4 = f(x) f( -x) = f(x) f is even.

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**Example 10 a) c) b) d) Continue…**

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

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**Example 11 Continue… Complete the graph of the following if**

b) Symmetric w.r.t origin a) Symmetric w.r.t y-axis 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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**Vertical Shifting Example 12 y = f( x ) + c Up c units y = f ( x ) - c**

Down c units Example 12 Below is the graph of a function y = f ( x ). Sketch the graphs of 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 Continued… y = f( x + c ) Left c units**

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

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**Left h units and Up k units Left h units and Down k units**

Example 14 Continued… 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 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 Example 15 Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1 y = f( x + 2 ) - 1 y = f( x )

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

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**Vertical Compress by a factor 1/c**

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 . Sketch the graphs of y = 5 x2 2. y = (1/5)x2 x y = x2 y=5x2 y=1/5x2 2 10 .4 1 5 .2 -1

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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. P ( 0, 5 ) y = f( x + 2 ) – 1 P ( 3, -1 ) y = 2f(x) +4 3) P( -2,4) y = (1/2) f( x-3) + 3 Solution: 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 ). 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) 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 Example 19 Below is the graph of a function y = x2 . Sketch the graph of y = - x2 x y = x2 y = -x2 2 4 -4 1 -1 -2

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**Sketching a piece-wise function**

Definition: Piece-wise function is a function that can be described in more than one expression. Example 20 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 = -1. The point (-1, 3 ) is included. Graph y = 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**

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. Example 21 Sketch the graph of y = g ( x ) = Solution: Strategy: 1. Graph y = f(x) = x2. 2. Graph y = f( x ) - 9 = x2 – 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 = x2 – 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 Solution: A picture can replace 1000 words**

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

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**Do all homework exercises in the syllabus**

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