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Advance Mathematics

Advance Mathematics. Section 3.5. Objectives: Define Even and Odd functions algebraically and graphically Sketch graphs of functions using shifting, and reflection. Even and Odd Functions. Example 9. Determine whether a function is even, odd or neither.

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Advance Mathematics

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

  2. Even and Odd Functions Example 9 Determine whether a function is even, odd or neither. a) f ( x ) = 3x4 + 5x2 –4 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.

  3. Example 10 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

  4. Example 11 Complete the graph of the following if b) Symmetric w.r.t origin Continue… 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

  5. Vertical Shifting 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

  6. Horizontal Shifting Continued… • 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

  7. Example 14 Can you tell the effects on the graph of y = f ( x ) Continued… Left h units and Up k units Lefth units and Down k units Righth 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 )

  8. 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

  9. Vertical Stretching 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

  10. 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 • 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 ). Solution:

  11. Reflecting a graphthrough the x-axis 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

  12. 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 = x2and 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.

  13. 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

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

  15. Do all homework exercises in the syllabus

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