Download Quiz 1-2

r 2  3r  54
2x2  4x  4
Lesson 1-6
Graphical Transformations
Graphical Transformations
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
(making the graph more steep or less steep)
Reflecting across x-axis or y-axis
Graphical Transformations
Parent Function: The simplest function in a family
of functions (lines, parabolas, cubic functions, etc.)
yx
2
y  x 2
2
Compare the two parabolas.
yx
y  x 2
2
2
Why does adding 2 to the
parent function shift the
graph up by 2?
Build a table of values for each equation for domain
elements: -2, -1, 0, 1, 2.
x
y
x
y
-2
4
1
0
1
4
-2
6
3
2
3
6
-1
0
1
2
-1
0
1
2
Your Turn:
Describe the transformation to the parent
function:
y  x 4
2
translated down 4
Describe the transformation to the parent
function:
y  x 5
2
yx
2
translated up 5
yx
2
Graphical Transformations
yx
y  3x
2
Compare the two parabolas.
Multiplying the parent
function by 3, makes it 3
times as steep.
2
yx
2
Why does multiplying the
parent function by 3
cause the parent to be
vertically stretched by a
factor of 3?.
y  3x 2
Build a table of values for each equation for domain
elements: -2, -1, 0, 1, 2.
x
-2
-1
0
1
2
y
4
1
0
1
4
x
y
-2
12
3
0
3
12
-1
0
1
2
Graphical Transformations
yx
2
y  x2
Multiplying the parent
function by -1, reflects
across the x-axis.
Compare the two parabolas.
Your Turn:
Describe the transformation to the parent
function:
y  x  2
2
Reflected across x-axis
and translated up 2
Describe the transformation to the parent
function:
y  3x  6
2
y  x2
Vertically stretched by
a factor of 3 and
translated down 6
yx
2
Graphical Transformations
yx
2
y  ( x  1)
Compare the two parabolas.
2
yx
y  ( x  1)
2
Why does replacing ‘x’
with ‘x – 1’ translates the
parent function right by 1.
Build a table of values for each equation for domain
elements: -2, -1, 0, 1, 2.
x
y
x
y
-2 9
-2 4
-1 4
-1 1
0
1
0
0
1 0
1
1
2 1
2
4
2
Quadratic Transformations
y  (1)a( x  h)  k
2
Reflection
across x-axis
vertical
stretch
factor
Translates
left/right
translating up
or down
y  2( x  3)  4
2
Reflected across x-axis, twice as steep,
translated up 4, translated right 3
Your Turn:
Describe the transformation to the parent
function:
y  ( x  5)  3
2
translated up 3
translated left 5
y  x2
Your Turn:
Describe the transformation to the parent
function:
2
y  2( x  1)
Vertically stretched by a factor of 2,
translated right 1
y  x2
Your Turn:
Describe the transformation to the parent
function:
1
2
y   ( x  3)  4
2
Reflected across x-axis
Vertically stretched by a factor of ½ (shrunk),
translated up 4
translated left 3
y  x2
Absolute Value Function
Why does it have this shape?
f ( x)  x
Your turn:
y x
What is the transformation to the parent function?
y  x 3
translated right 3
y 2x
Vertically stretched
by a factor of 2 
Twice as steep
Slope on right side is +2
slope on left side is -2
Your turn:
y x
What is the transformation to the parent function?
y  3 x  2  4
Reflected across x-axis
VSF = 4  4 times as steep
Left 2
up 4
Absolute Value Transformation
y  (1)a x  h  k
Reflection
across x-axis
Vertical
stretch
factor
Translates
left/right
translating up
or down
What does adding or subtraction “k” do to the parent
function?
f ( x)  x  k
Vertical shift
What does adding or subtraction “h” do to the parent
function?
f ( x)  x  h
Horizontal shift
What does multiplying by ‘a’ do to the parent
function?
f ( x)  a x
Vertical stretch
What does multiplying by (-1) do to the parent
function?
f ( x)   x
Reflection across x-axis
Square Root Function
What is the domain of the graph?
f ( x)  x
Describe the transformation to the parent function:
y  4 x2
y  3  2 x  3
Up 4, right 2
y  x2 4
Down 3,
reflected across x-axis,
VSF=2
left 3
y  (1)a x  h  k
Reflecting Across the x-axis
Reflecting across the y-axis
example
Question:
• What happens when an even function is
reflected across the y-axis?
Homework:
• HW 1-6 pg 147: 2-32 Even