2.2 reflections & transformations. we need to have a firm grasp on our basic graphs, so we can...
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2.2 Reflections & Transformations
We need to have a firm grasp on our basic graphs, so we can then easily transform them using some simple rules.
6 Basic Graphs:
(1) Identity (2) Squaring( )f x x (3) Cubing
(4) Square Root
(5) Absolute Value
(6) Reciprocal
y x
2( )f x x
2y x
3( )f x x
( )f x x ( )f x x1
( )f xx
3y x
y x y x
1y
x
3 ways to change up a graph – Reflections, Translations, & DilationStart with y = f (x)
Reflections
y = – f (x) y = f (– x) y = f –1 (x)
Reflection in x-axis Reflection in y-axis Reflection in y = x(inverse!)
Translations
Horizontal Shift Vertical Shift
(since it is –c, think the opposite) (x – c) c to the right (x + c) c to the left
take it for what it is +d d up –d d down
shift y = f (x – c) + d
Dilations
a > 1 0 < a < 1
Taller or skinnier(away from x-axis)
Wider or fatter(towards x-axis)
stretch or compress y = af (x)
Let’s Explore! iPad app Desmos (can also access online @ desmos.com )
Start Desmos Find button in top left cornerChoose transformations Reflections of a functionExplore this graph!
touch next to the 5 & delete up to =enter your ownchange box 4 to x = f (y) (use ABC)touching next to the next to the formula will “turn on” each graphAre they what you expected?
x
Launch website
Now let’s explore translationsGo to , under Transformations Translating Any Function delete function in box 1go to put x in ( )
Play with h & k put in negative values/ positive valuesuse slider or enter a # in boxes 5 & 7 or change values in equation directly in box 3
Launch website
Time for DilationsGo to , under Transformations Scaling Any Function delete function in box 1type in x2
Play with a make bigger or smaller – use slider or type in #s
Bring it all together!Describe (yes, write a sentence!) how the basic graph of y = x3 has been changedThen check on Desmos or calculatorTouch & Type in the equation & confirm your description!
y = 2 (x + 3)3 – 2
Launch website
Graph
Homework
#202 Pg 69 #1–43 odd, 38, 40, 42, 44