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# 2.5 translations of graphs

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## 2.5 translations of graphsPresentation Transcript

• Translations of Graphs
• Translations of Graphs
Objective:
* Vertical stretches and
compressions of graphs
* Vertical and horizontal translations
of graphs
• Translations of Graphs
Stretch and Compress
• Translations of Graphs
Stretch and Compress
The graph of y=c*f(x) is a
vertical stretch or compression
of the graph y = f(x).
• Translations of Graphs
Stretch and Compress
The graph of y=c*f(x) is a
vertical stretch or compression
of the graph y = f(x).
• Translations of Graphs
Stretch and Compress
The graph of y=c*f(x) is a
vertical stretch or compression
of the graph y = f(x).
Stretch (2x-x2) vertically by a factor of 2
y=2(2x-x2)
 Frank Ma
2006
• Translations of Graphs
Stretch and Compress
The graph of y=c*f(x) is a
vertical stretch or compression
of the graph y = f(x).
If c > 1, stretch the graph vertically by a factor of c
y=c(2x-x2)
 Frank Ma
2006
• Translations of Graphs
Stretch and Compress
The graph of y=c*f(x) is a
vertical stretch or compression
of the graph y = f(x).
y=c(2x-x2)
If 1 > c > 0,compress vertically by the factor of c
y=0.5(2x-x2)
• Translations of Graphs
Flip across the x-axis if c<0.
• Translations of Graphs
Flip across the x-axis if c<0.
• Translations of Graphs
Flip across the x-axis if c<0.
y= – (2x-x2)
• Translations of Graphs
Flip across the x-axis if c<0.
 Frank Ma
2006
• Translations of Graphs
Stretch and Compress
Given the graph of y=f(x), the graph of y=c*f(x) is the vertical-stretch or compression of y=f(x).
 Frank Ma
2006
• Translations of Graphs
Stretch and Compress
Given the graph of y=f(x), the graph of y=c*f(x) is the vertical-stretch or compression of y=f(x).
If c>1, it is a vertical stretch.
If 0<c<1, it is a vertical compression.
 Frank Ma
2006
• Translations of Graphs
Stretch and Compress
Given the graph of y=f(x), the graph of y=c*f(x) is the vertical-stretch or compression of y=f(x).
If c>1, it is a vertical stretch.
If 0<c<1, it is a vertical compression.
If c is negative (c<0), it flips vertically
across the x-axis.
 Frank Ma
2006
• Translations of Graphs
Vertical Translations
The graph of y=f(x)+c is a vertical translation of y=f(x).
• Translations of Graphs
Vertical Translations
The graph of y=f(x)+c is a vertical translation of y=f(x).
y=x2+5
y=x2
5
• Translations of Graphs
Vertical Translations
The graph of y=f(x)+c is a vertical translation of y=f(x).
y=x2+5
y=x2
y=x2–5
5
-5
• Translations of Graphs
Vertical Translations
The graph of y=f(x)+c is a vertical translation of y=f(x).
y=x2+5
y=x2
y=x2–5
5
-5
If c>0, the graph moves up.
If c<0, the graph moves down.
• Translations of Graphs
Vertical Translations
The graph of y=f(x)+c is a vertical translation of y=f(x).
If c>0, the graph moves up.
If c<0, the graph moves down.
• Horizontal Translations
• Horizontal Translations
The graphs of y=f(x±c) are the horizontal translations of y=f(x) by c units.
• Horizontal Translations
The graphs of y=f(x±c) are the horizontal translations of y=f(x) by c units.
For y=f(x + c), moves y=f(x) to the left. (c>0)
• Horizontal Translations
The graphs of y=f(x±c) are the horizontal translations of y=f(x) by c units.
For y=f(x + c), moves y=f(x) to the left. (c>0)
For y=f(x – c), moves y=f(x) to the right. (c>0)
• Horizontal Translations
The graphs of y=f(x±c) are the horizontal translations of y=f(x) by c units.
For y=f(x + c), moves y=f(x) to the left. (c>0)
For y=f(x – c), moves y=f(x) to the right. (c>0)
• Horizontal Translations
The graphs of y=f(x±c) are the horizontal translations of y=f(x) by c units.
For y=f(x + c), moves y=f(x) to the left. (c>0)
For y=f(x – c), moves y=f(x) to the right. (c>0)
y=(x – 3)2
• Horizontal Translations
The graphs of y=f(x±c) are the horizontal translations of y=f(x) by c units.
For y=f(x + c), moves y=f(x) to the left. (c>0)
For y=f(x – c), moves y=f(x) to the right. (c>0)
y=(x + 3)2
y=(x – 3)2
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)  (-2, 1)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)  (0, 2)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)  (0, 2)  (0, 5)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)  (0, 2)  (0, 5)
(1, 1)  (2, -2)  (2, 1)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)  (0, 2)  (0, 5)
(1, 1)  (2, -2)  (2, 1)
(2, 1)  (3, -2)  (3, 1)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(0, 5)
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)  (0, 2)  (0, 5)
(-2, 1)
(3, 1)
(2, 1)
(1, 1)  (2, -2)  (2, 1)
(2, 1)  (3, -2)  (3, 1)
• Absolute-Value Flip
(-3, 1)
(1, 1)
(2, 1)
Example A:
Given the graph of y = f(x),
graph y = -2f(x – 1) + 3
(-1, -1)
"-2f(x – 1)" corresponds moving the graph 1 unit to the right and stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3" corresponds to moving the graph vertically by 3.
A good way to do this is to shift the important points on the graph.
(0, 5)
(-3, 1)  (-2, -2)  (-2, 1)
(-1, -1)  (0, 2)  (0, 5)
(-2, 1)
(3, 1)
(2, 1)
(1, 1)  (2, -2)  (2, 1)
(2, 1)  (3, -2)  (3, 1)
• Absolute-Value Flip
• Absolute-Value Flip
y = f(x) = x
• Absolute-Value Flip
y = f(x) = x
• Absolute-Value Flip
y = |f(x)| = |x|
y = f(x) = x
• Absolute-Value Flip
y = |f(x)| = |x|
y = f(x) = x
• Absolute-Value Flip
y = |f(x)| = |x|
y = f(x) = x
The graph of y = |f(x)| is obtained by reflecting the portion of the graph below the x-axis to above the x-axis.
• Absolute-Value Flip
Another example,
y = x2 – 1
• Absolute-Value Flip
Another example,
y = |x2 – 1|
y = x2 – 1
• Absolute-Value Flip
Another example,
y = |x2 – 1|
y = x2 – 1
• Absolute-Value Flip
Another example,
y = |x2 – 1|
y = x2 – 1
y = |x2 – 1| – 1
• Absolute-Value Flip
Another example,
y = |x2 – 1|
y = x2 – 1
y = |x2 – 1| – 1
• Absolute-Value Flip
Another example,
y = |x2 – 1|
y = x2 – 1
y = |x2 – 1| – 1
y = 2(|x2 – 1| – 1)
• Absolute-Value Flip
Another example,
y = |x2 – 1|
y = x2 – 1
y = |x2 – 1| – 1
y = 2(|x2 – 1| – 1)
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
y = f(x) = x3 – x2
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
y = f(-x) = (-x)3 – (-x)2
y = f(x) = x3 – x2
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
y = f(-x) = (-x)3 – (-x)2
y = f(x) = x3 – x2
y = f(-x) = – x3 – x2
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
y = f(-x) = (-x)3 – (-x)2
y = f(x) = x3 – x2
y = f(-x) = – x3 – x2
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
y = f(-x) = (-x)3 – (-x)2
y = f(x) = x3 – x2
y = f(-x) = – x3 – x2
A function is said to be even if f(x) = f(– x). Graphs of even functions are symmetric to the
y-axis.
• Horizontal Flip
The graph of y = f(–x) is the horizontal reflection of
the graph of y = f(x) across the y axis.
y = f(-x) = (-x)3 – (-x)2
y = f(x) = x3 – x2
y = f(-x) = – x3 – x2
A function is said to be even if f(x) = f(– x). Graphs of even functions are symmetric to the
y-axis.
Graph of an even function
• Horizontal Flip
Polynomial-functions whose terms are all even power are even.
• Horizontal Flip
Polynomial-functions whose terms are all even power are even. The graph on the right is y = x4 – 4x2.
y = x4 – 4x
• Horizontal Flip
Polynomial-functions whose terms are all even power are even. The graph on the right is y = x4 – 4x2.
y = x4 – 4x
A function is said to be odd iff f(x) = – f(– x).
• Horizontal Flip
Polynomial-functions whose terms are all even power are even. The graph on the right is y = x4 – 4x2.
y = x4 – 4x
A function is said to be odd iff f(x) = – f(– x).
Graphs of odd functions are symmetric to the origin,
• Horizontal Flip
Polynomial-functions whose terms are all even power are even. The graph on the right is y = x4 – 4x2.
y = x4 – 4x
A function is said to be odd iff f(x) = – f(– x).
Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis.
• Horizontal Flip
Polynomial-functions whose terms are all even power are even. The graph on the right is y = x4 – 4x2.
y = x4 – 4x
A function is said to be odd iff f(x) = – f(– x).
Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis.
Graph of an odd function
• Horizontal Flip
Polynomial-functions whose terms are all even power are even. The graph on the right is y = x4 – 4x2.
y = x4 – 4x
A function is said to be odd iff f(x) = – f(– x).
Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis.
Graph of an odd function
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):)
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
The product of an even function with an odd
function is odd.
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
The product of an even function with an odd
function is odd. (The same hold for quotient.)
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
The product of an even function with an odd
function is odd. (The same hold for quotient.)
x
is odd,
x4 + 1
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
The product of an even function with an odd
function is odd. (The same hold for quotient.)
x
x2
is odd, is even,
x4 + 1
x4 + 1
• Horizontal Flip
Polynomial-functions whose terms are all odd power are odd.
The graph on the right is y = x3 – 4x.
y = x3 – 4x
Theorem (even and odd):
I. The sum of even functions is even.
The sum of odd functions is odd.
II. The product of even functions is even.
The product of odd functions is even.
The product of an even function with an odd
function is odd. (The same hold for quotient.)
x
x2
is odd, is even, x + 1 is neither.
x4 + 1
x4 + 1
• HW
Use the graph of y=x2, sketch the
graphs of:
1. y = 3x2
2. y = -2x2
3. y = -0.5x2
4. y=x2 – 1
5. y=2x2 – 1
6. y= -x2 – 2
7. y=(x+1)2
8. y=(x–3)2
 Frank Ma
2006