1.7 Transformations of Functions
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Transcript of 1.7 Transformations of Functions
• Pass out student note handouts
On graph paper, graph the following functions
2)()( xxf
2)( 2 xxg 2)4()( xxk2)2()( xxl 22)( xxh
https://www.desmos.com/calculator
1.7 Transformations of Functions
2)( xxj
2)( xxf
I. There are 4 basic transformations for a function f(x).y = A • f (Bx + C) + D
A) f(x) + D (moves the graph + ↑ and – ↓)
B) A • f(x)
1) If | A | > 1 then it is vertically stretched.
2) If 0 < | A | < 1, then it’s a vertical shrink.
3) If A is negative, then it flips over the x-axis.
C) f(x + C) (moves the graph + and – )
D) f(Bx) or f(B(x)) (factor out the B term if possible)
1) If | B | > 1 then it’s a horizontal shrink.
2) If 0 < | B | < 1, then it’s horizontally stretched.
3) If B is negative, then it flips over the y-axis.
Attached to the y – vertical and intuitive
Attached to the x – horizontal and counter-intuitive
1.7 Transformations of Functions
II. What each transformation does to the graph.
A) f(x) f(x) + D f(x) – D
B) +A f(x) +A f(x) –A f(x) . A > 1 0 < A < 1
1.7 Transformations of Functions
II. What each transformation does to the graph.
C) f(x) f(x + C) f(x – C)
D) f(Bx) f(Bx) f(-Bx) . B > 1 0 < B < 1
1.7 Transformations of Functions
III. What happens to the ordered pair (x , y) for shifts.
A) f(x) + D (add the D term to the y value)
Example: f(x) + 2 (5 , 4)
f(x) – 3 (5 , 4)
B) A • f(x) (multiply the y value by A)
Example: 3 f(x) (5 , 4)
½ f(x) (5 , 4)
–2 f(x) (5 , 4)
C) f(x + C) (add –C to the x value) [change C’s sign]
Example: f(x + 2) (5 , 4) (subtract 2)
f(x – 3) (5 , 4) (add 3)
)6,5()1,5(
)12,5()2,5()8,5(
)4,3()4,8(
1.7 Transformations of Functions
III. What happens to the ordered pair (x , y) for shifts.
D) f(Bx) or f (B(x))
1) If B > 1 (divide the x value by B)
Example: f(2x) (12 , 4)
f(3x) (12 , 4)
f (4(x)) (12 , 4)
2) If 0<B<1 (divide the x value by B) [flip & multiply]
Example: f(½x) (12 , 4) f (¾(x)) (12 , 4)
3) If B is negative (follow the above rules for dividing)
Example: f(-2x) (12 , 4) f (-½(x)) (12 , 4)
)4,6()4,4()4,3(
)4,24()4,16(
)4,6()4,24(
1.7 Transformations of Functionsf(x) is shown below. Find the coordinates for the following shifts.
f(x) + 4 f(x) – 6
2 f(x) ½ f(x) -3 f(x)
f(x + 4) f(x – 3)
f(2x) f(½x) f(-3(x))
(-4,6) (-1,4)
(1,7 ) (2,1)
(-4,-4) (-1,-6)
(1,-3) (2,-9)
(-8,2) (-5,0)
(-3,3) (-2,-3)
(-1,2) (2,0)
(4,3) (5,-3)
(-4,4) (-1,0)
(1,6) (2,-6)
(-4,1) (-1,0)
(1,3/2) (2,-3/2)
(-4,-6) (-1,0)
(1,-9) (2,9)
(-2,2) (-1/2,0)
(1/2,3) (1,-3)
(-8,2) (-2,0)
(2,3) (4,-3)
(4/3,2) (1/3,0)
(-1/3,3) (-2/3,-3)
• Identify the parent function and describe the sequence of transformations.
1.7 Transformations of Functions
1)(
)8()(
3
2
xxh
xxg2)( xxf Horizontal shift eight units
to the right
3)( xxf Reflection in the x-axis, and a vertical shift of one unit downward
or y-axis!
• Identify the parent function and describe the sequence of transformations.
• Parent Function
• Left 2
• Horizontally compressed by a factor of 1/2
2)42()( xxk
1.7 Transformations of Functions
2)]2(2[ x2)( xxf
Always factor
If possible!
• Identify the parent function and describe the sequence of transformations.
• Flip over y-axis and right 4
• If x is negated, factor out a negative!
2)4()( xxk
1.7 Transformations of Functions
2)]4([ x
• When graphing, perform non-rigid transformations 1st and rigid transformations last
• That means stretch / compress / reflect before moving left / right / up / down
• Then find a few points and perform transformations on those points.
• Ex: Graph
• Ex: Graph
1)2(2
1)( 2 xxf
3)2()( 3 xxf
Practice
• Ex: Graph
• Ex: Graph
4)3(2)( 2 xxf
2)42()( 3 xxf
H Dub
• 1-7 Page 80 #9-12 (parts A and B only), 13-18all, 19-39EOO