Upload
vuongphuc
View
217
Download
0
Embed Size (px)
Citation preview
Vocabulary Transformation: an adjustment made to the parent function
that results in a change to the graph of the parent function.
Changes could include:
shifting (“translating”) the graph up or down,
“translating” the graph left or right
vertical stretching or shrinking
Reflecting across x-axis or y-axis
horizontal stretching or shrinking
Graphical Transformations Parent Function: The simplest function in a family
of functions (lines, parabolas, cubic functions, etc.)
2xy
2xy
Replacing ‘y’ with y – 2 results in an up two translation.
22 xy
22 xyWe usually write
“y as a function of x”
“y” on one side of “=“ sign,
operations performed on ‘x’ on the other side of the “=“ sign
Adding ‘2’ to the parent function translates the parent
function up two.
2)( 2 xxg2)( xxf
Build a table of values for each equation for domain
elements: -2, -1, 0, 1, 2.
Why does adding 2 to the
parent function translate
the graph up by 2?
x f(x)
-2
-1
0
1
2
4
1
0
1
4
x g(x)
-2
-1
0
1
2
6
3
2
3
6
For each respective
input value, the
output value has
increased by 2.
22
2 xy2
1 xy Notice the subscripts.
Why did I put different
subscripts on the two
equations?
shows they are different equations.
easier to see in function form with different names 2
1 xy 2)( xxf
22
2 xy
2)( 2 xxg
2)()( xfxg
Take Away: add 2 to the parent function move up 2
Your Turn: Describe the transformation to the parent
function:
2xy
42 xy
Describe the transformation to the parent
function:
2xy
52 xy
translated down 4
translated up 5
2)( xxf Why does multiplying
the parent function by 3
cause the parent to
look steeper?
23)( xxg
Build a table of values for each equation for some of
the input values: -2, -1, 0, 1, 2.
x f(x)
-2
-1
0
1
2
x g(x)
-2
-1
0
1
2
4
1
0
1
4
12
3
0
3
12
Same input value
output value has
been multiplied by 3.
We say the function
has been “vertically
stretched” by a
factor of 3.
f(x) is given in
the table below
x f(x)
-2 -2
-1 0
0 2
1 4
2 6
g(x) = 3*f(x)
Fill in the table below
x g(x)
-2
-1
0
1
2
-6
0
6
12
18
2)( xxf
Multiplying the x-value by 2,
horizontally shrinks the graph.
2)2()( xxg 2)2()2()( xxfxg
x f(x)
-2
-1
0
1
2
4
1
0
1
4
x f(2x)
-2
-1
0
1
2
Looks like a vertical stretch!
16
4
0
4
16
22 4)2()( xxxg
For the square function:
Horizontal shrinking
Looks like
Vertical stretching.
2)( xxf 22 4)2()( xxxg
For the square function:
Horizontal stretch by ½
(multiply x-value of point by ½
Looks like
Vertical stretch by 4
(multiply y-value of point by 4).
2)( xxf Multiplying the parent
function by -1, reflects
across the x-axis.
2)( xxg
2)()( xxfxg
x f(x)
-2
-1
0
1
2
4
1
0
1
4
x -f(x)
-2
-1
0
1
2
Multiplying the parent
function by -1, multiplies
each y-value by -1.
-4
-1
0
-1
-4
2)( xxf 2)1()( xxg
Remember? Replacing ‘y’ with ‘y-1’ moved it up 1.
Replacing ‘x’ with ‘x-1’ moves it right 1.
2)( xxf
Build a table of values for each equation for domain
elements: -2, -1, 0, 1, 2.
x f(x)
-2
-1
0
1
2
x g(x)
-2
-1
0
1
2
4
1
0
1
4
9
4
1
0
1
2)1()( xxg
)()1()1( 2 xgxxf
Replacing ‘x’ in the original function with ‘x – 1’
causes the graph to translate right ‘1’
These effects accumulate
Describe the transformation to the parent
function:
2)( xxf
2)( 2 xxg
2)( 2 xxg
2)()( xfxg
Describe algebraically how f(x) is transformed to get g(x).
2)( xxf
)1(* )1(* 2)( xxf
2 2
22)( 2 xxf
These effects accumulate
Describe the transformation to the parent
function:
2)( xxf
2)( 2 xxg
2)( 2 xxg
Reflected across x-axis and translated up 2
2)()( xfxg
Describe graphically how f(x) is transformed to get g(x).
These effects accumulate
2)( xxf
Describe the algebraic transformation to the
parent function:
Multiplying the parent function by 3 then subtracting 6…
63)( 2 xxg6)(3)( xfxg
2)( xxf
3* 3*
23)(3 xxf
6 6
636)(3 2 xxf
These effects accumulate
2)( xxf
Describe the graphical transformation to the
parent function:
Multiplying the parent function by 3 then
subtracting 6…
Vertically stretched by a factor of 3 and
translated down 6
63)( 2 xxg
6)(3)( xfxg
Let’s generalize the transformations
Reflection
across x-axis
translating
up or down vertical
stretch
factor
khxay 2)()1(
Translates
left/right
4)3(2 2 xy
Reflected across x-axis, twice as steep,
translated up 4, translated right 3
2)( xxf
Your Turn:
Describe the transformation to the parent
function: 2)( xxf 3)5( 2 xy
translated up 3
translated left 5
3)5(3)5( 22 xxf
Your Turn:
Describe the transformation to the parent
function: 2xy 2)1(2 xy
Vertically stretched by a
factor of 2, translated right 1
Your Turn:
Describe the transformation to the parent
function: 2xy 4)3(2
1 2 xy
Reflected across x-axis
Vertically stretched by a factor of ½
(shrunk), translated up 4
translated left 3
Your Turn:
Describe the transformation to the parent
function: 2xy 5)2( 2 xy
Horizontally shrunk by 2 (or stretched by ½)
translated up 5