• 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