Presentation on theme: "Advance Mathematics Section 3.5 Objectives:"— Presentation transcript:
1 Advance Mathematics Section 3.5 Objectives: Define Even and Odd functions algebraically and graphicallySketch graphs of functions using shifting, and reflection
2 f(-x) = -2 (-x)5 + 4( -x )3 +7(-x) f(-x) = ( -x )5 + ( -x )2 Even and Odd FunctionsTerminologyDefinitionExampleType of Symmetryf is aneven functionf(-x) = f(x)y = f(x) = x2w.r.ty-axisodd functionf(-x) = - f(x)y = f(x ) = x3OriginExample 9Determine whether a function is even, odd or neither.a) f ( x ) = 3x4 + 5x2 – b) f( x ) = -2x5 +4x3 +7x c) f( x ) = x3 +x2Substitute x by –xf(-x) = -2 (-x)5 + 4( -x )3 +7(-x)= 2x5 – 4x3 – 7x= - (-2x5 +4x3 +7 )= - f(x) f(-x) = - f(x)f is oddSolution:Substitute x by –xf(-x) = ( -x )5 + ( -x )2= - x5 + x2f(-x) is not equal to f( x) nor –f(x). Therefore, f is neither.Substitute x by –xf( -x) = 3( -x )4 + 5 ( -x )2 - 4= 3x4 + 5x2 – 4= f(x)f( -x) = f(x)f is even.
3 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 functionThe graph represents neitherb)d)The graph represents neitherThe graph represents an odd function
4 Example 11 Continue… Complete the graph of the following if b) Symmetric w.r.t origina) Symmetric w.r.t y-axisc) Function is evend) Function is odde) Symmetric w.r.t x-axisGraph of (e ) does not represent a function
5 Vertical Shifting Example 12 y = f( x ) + c Up c units y = f ( x ) - c Down c unitsExample 12Below is the graph of a function y = f ( x ). Sketch the graphs ofy = f ( x ) + 1b) y = f ( x ) - 2y = f(x) + 1y = f (x)y = f (x)-2
6 Horizontal Shifting Continued… y = f( x + c ) Left c units Right c unitsExample 13.Given the graph of a functiony = f ( x ). Sketch the graphs ofy = f ( x + 3 )b) y = f ( x – 4 )y = f ( x )y = f ( x ) + 3Horizontal Shift 3 units to the lefty = f ( x )y = f ( x ) - 4Horizontal Shift 4 units to the right
7 Left h units and Up k units Left h units and Down k units Example 14Continued…Can you tell the effects on the graph of y = f ( x )y = f( x + h ) + ky = f( x + h ) - ky = f( x - h ) + ky = f( x - h ) - kLeft h units and Up k unitsLeft h units and Down k unitsRight h units and Up k unitsRight h units and Down k unitsExample 15Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1y = f( x + 2 ) - 1y = f( x )
8 Example 16Continued…Below is the graph of a function Sketch the graph ofSolution:The graph of the absolute value is shifted 2 units to the right and 3 units downy = f( x )y =f(x-2)-3
9 Vertical Compress by a factor 1/c Vertical Stretchingy = cf( x) ( c> 1 )Vertical Stretch by afactor cy = (1/c)f ( x) ( c > 1 )Vertical Compress by a factor 1/cNote1 :When c > 1. Then 0 < 1/c < 1Note 2 : c effects the value of y only.Example 17Below is the graph of a function y = x2 . Sketch the graphs ofy = 5 x22. y = (1/5)x2xy = x2y=5x2y=1/5x2210.415.2-1
10 Example 18If 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 ) – 1P ( 3, -1 ) y = 2f(x) +43) P( -2,4) y = (1/2) f( x-3) + 3Solution: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 ofy = (1/2)(4) + 3 = 5. Therefore, the corresponding point is ( 1, 5 ).
11 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 - yExample 19Below is the graph of a function y = x2 . Sketch the graph ofy = - x2xy = x2y = -x224-41-1-2
12 Sketching a piece-wise function Definition: Piece-wise function is a function that can be described in more than one expression.Example 20Sketch the graph of the function f ifSolution: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 includedGraph 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 functionWe have to remember thatAnd 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 21Sketch 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 down3. 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 Solution: A picture can replace 1000 words Below is the graph of y = f(x). GraphSolution:A picture can replace 1000 wordsLet the animation talk about itself