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this time (f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) (fg)(x)=f(x)*g(x) (f/g)(x)=f(x)/g(x), g(x)≠0 (f...

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  • This time (f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) (fg)(x)=f(x)*g(x) (f/g)(x)=f(x)/g(x), g(x)≠0 (f ∘ g)(x)=f(g(x))
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This time • (f+g)(x)=f(x)+g(x) • (f-g)(x)=f(x)-g(x) • (fg)(x)=f(x)*g(x) • (f/g)(x)=f(x)/g(x), g(x)≠0 • (f∘g)(x)=f(g(x))

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  • Slide 1
  • This time (f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) (fg)(x)=f(x)*g(x) (f/g)(x)=f(x)/g(x), g(x)0 (f g)(x)=f(g(x))
  • Slide 2
  • Things to remember Function notation (x)=2x-1 is a function definition x is a number (x) is a number 2x-1 is a number is the action taken to get from x to (x) Multiply by 2 and add -1
  • Slide 3
  • Things to remember Function notation (x)=2x-1 is a function definition 3 is a number (3) is a number 2*3-1 is a number (its 5) is the action taken to get from 3 to (3) Multiply by 2 and add -1
  • Slide 4
  • Practice using notation
  • Slide 5
  • I can do a function to a number
  • Slide 6
  • Whats going on here?
  • Slide 7
  • Composing functions algebraically
  • Slide 8
  • In diagram form
  • Slide 9
  • WARNING Parentheses are ambiguous When you have two NUMBERS, a(b) means multiply a and b When you have a FUNCTION, a(b) means do the action called a to the number b. Always keep track of whats a function and whats a number.
  • Slide 10
  • The most common confusion of all time (f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) (fg)(x)=f(x)*g(x) (f/g)(x)=f(x)/g(x), g(x)0 (f g)(x)=f(g(x))
  • Slide 11
  • COMPARISON (f g)(3)=f(g(3)) (fg)(3)=f(3)g(3) -10
  • Slide 12
  • WARNING (fg)(x) and f(g(x)) are not the same thing (fg)(x) means do f to x, then do g to x, then multiply the numbers f(x) and g(x). f(g(x)) means do g to x, get the number g(x), then do f to the number g(x) No multiplying.
  • Slide 13
  • In picture form x f(x) g(x) f(x)g(x) * f g xg(x)f(g(x)) fg Is not the same as
  • Slide 14
  • Interpretive Dance All about multiplication
  • Slide 15
  • Time to dance!
  • Slide 16
  • Add 1 to each number
  • Slide 17
  • Add -1 to each number
  • Slide 18
  • Multiply each number by 0.5
  • Slide 19
  • Multiply each number by 2
  • Slide 20
  • Multiplication is not repeated addition Addition is shifting Multiplication is stretching And shrinking No amount of repeated shifting will give you a stretch
  • Slide 21
  • Transformations of functions And their graphs
  • Slide 22
  • This is a graph of a function called
  • Slide 23
  • Let g(x)=(x)+1.5
  • Slide 24
  • What does a graph of g look like?
  • Slide 25
  • g(3)=(3)+1.5 (3,f(3)) (3,f(3)+1.5)
  • Slide 26
  • g(3)=(3)+1.5 (3,f(3)) (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher The x is still the same, But the y is 1.5 higher
  • Slide 27
  • Draw a graph where all the xs are the same and all the ys are 1.5 higher. (3,f(3)) (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher The x is still the same, But the y is 1.5 higher
  • Slide 28
  • The graph of f(x)+1.5 is the graph of f(x) shifted up by 1.5 (3,f(3)) (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher The x is still the same, But the y is 1.5 higher
  • Slide 29
  • Draw a graph where all the xs are the same and all the ys are 1.5 higher. (3,f(3)) (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher The x is still the same, But the y is 1.5 higher
  • Slide 30
  • Graphing Transformations The graph for (x)+c is the graph of (x) shifted up by c.
  • Slide 31
  • This is a graph of a function called
  • Slide 32
  • Let g(x)=(x-1)
  • Slide 33
  • What does a graph of g look like?
  • Slide 34
  • g(3)=(3-1)=(2) (2,f(2))(3,f(2))
  • Slide 35
  • g(3)=(3-1)=(2) (2,f(2))(3,f(2)) The y of g(3) is the same as the y of f(2)
  • Slide 36
  • g(3)=(3-1)=(2) (2,f(2))(3,f(2)) The y of g(3) is the same as the y of f(2) Thinking from g, The y is the same, but the x needed to make that y is 1 bigger Thinking from g, The y is the same, but the x needed to make that y is 1 bigger
  • Slide 37
  • Draw a graph where the ys are the same, but you need a 1 bigger x to make each one. (2,f(2))(3,f(2)) The y of g(3) is the same as the y of f(2) Thinking from g, The y is the same, but the x needed to make that y is 1 bigger Thinking from g, The y is the same, but the x needed to make that y is 1 bigger
  • Slide 38
  • Graphing Transformations The graph for (x)+c is the graph of (x) shifted up by c. The graph for (x-a) is the graph of (x) shifted right by a. NOTE THE MINUS SIGN
  • Slide 39
  • This is a graph of a function called
  • Slide 40
  • Let g(x)=2(x)
  • Slide 41
  • g(-1.5)=2(-1.5) (-1.5,f(-1.5)) (-1.5,2f(-1.5))
  • Slide 42
  • g(-1.5)=2(-1.5) (-1.5,f(-1.5)) (-1.5,2f(-1.5)) The x is still the same, But the y is twice as far away from zero The x is still the same, But the y is twice as far away from zero
  • Slide 43
  • Draw a graph where the xs stay the same, but the ys are twice as far away from zero. (-1.5,f(-1.5)) (-1.5,2f(-1.5)) The x is still the same, But the y is twice as far away from zero The x is still the same, But the y is twice as far away from zero
  • Slide 44
  • Graphing Transformations The graph for (x)+c is the graph of (x) shifted up by c. The graph for (x-a) is the graph of (x) shifted right by a. NOTE THE MINUS SIGN The graph for r(x) is the graph of (x) stretched vertically by r.
  • Slide 45
  • This is a graph of a function called
  • Slide 46
  • Let g(x)=(2x)
  • Slide 47
  • g(1.5)=(2*1.5)=(3) (3,f(3)) (1.5,f(3)) The y is still the same, but the x needed to make that y is half as big
  • Slide 48
  • Draw a graph where the ys stay the same, but the xs needed to make those graphs are half as big. (3,f(3)) (1.5,f(3)) The y is still the same, but the x needed to make that y is half as big
  • Slide 49
  • Graphing Transformations The graph for (x)+c is the graph of (x) shifted up by c. The graph for (x-a) is the graph of (x) shifted right by a. NOTE THE MINUS SIGN The graph for r(x) is the graph of (x) stretched vertically by r. The graph for (sx) is the graph of (x) squished horizontally by s.
  • Slide 50
  • Graphing Transformations The graph for (x)+c is the graph of (x) shifted up by c. The graph for (x-a) is the graph of (x) shifted right by a. NOTE THE MINUS SIGN The graph for r(x) is the graph of (x) stretched vertically by r. The graph for (sx) is the graph of (x) squished horizontally by s. Note the difference!
  • Slide 51
  • What does it mean to stretch by a fraction? g(x)=0.5f(x)
  • Slide 52
  • What does it mean to stretch by a negative?
  • Slide 53
  • Graphing Transformations The graph for (x)+c is the graph of (x) shifted up by c. The graph for (x-a) is the graph of (x) shifted right by a. NOTE THE MINUS SIGN The graph for r(x) is the graph of (x) stretched vertically by r. negative r causes the graph to flip vertically. The graph for (sx) is the graph of (x) squished horizontally by s. Negative s causes the graph to flip horizontally Note the difference!
  • Slide 54
  • Even Function A function where a horizontal flip does not change the graph
  • Slide 55
  • Even Function A function where a horizontal flip does not change the graph Graph of the function f(x) Graph of the horizontal flip f(-x)
  • Slide 56
  • Even Function A function where a horizontal flip does not change the graph Graph of the function f(x) Graph of the horizontal flip f(-x) Even Function f(x)=f(-x)
  • Slide 57
  • Even Function A function where a horizontal flip does not change the graph Graph of the function f(x) Graph of the horizontal flip f(-x) Even Function f(x)=f(-x) Example: f(x)=x 2 f(-x)=(-x) 2 =x 2 =f(x)
  • Slide 58
  • Odd Function A function where a horizontal flip has same graph as a vertical flip.
  • Slide 59
  • Odd Function A function where a horizontal flip has same graph as a vertical flip. Function: f(x) Horizontal Flip: f(-x) Vertical Flip: -f(x) Odd function f(-x)=-f(x)
  • Slide 60
  • Odd Function A function where a horizontal flip has same graph as a vertical flip. Function: f(x) Horizontal Flip: f(-x) Vertical Flip: -f(x) Odd function f(-x)=-f(x) Example f(x)=x 3 f(-x)=(-x) 3 =-(x 3 )=-f(x)
  • Slide 61
  • Write an equation for a function that has the graph of x 2 but is shifted right 3 units and up 4 units. a)(x-3) 2 +4 b)(x-3) 2 -4 c)(x+3) 2 +4 d)(x+3) 2 -4 e)None of the above
  • Slide 62
  • Write an equation for a function that has the graph of x 2 but is shifted right 3 units and up 4 units. a)(x-3) 2 +4 b)(x-3) 2 -4 c)(x+3) 2 +4 d)(x+3) 2 -4 e)None of the above
  • Slide 63
  • Is f(x)=x 2 +3x-4 a)Even, not odd b)Odd, not even c)Both even and odd d)Neither even nor odd
  • Slide 64
  • Is f(x)=x 2 +3x-4 a)Even, not odd b)Odd, not even c)Both even and odd d)Neither even nor odd f(-x)=(-x) 2 -3x-4=x 2 -3x-4 f(-x)f(x) NOT EVEN F(-x)-f(x) NOT ODD

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