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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