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2.5 translations of graphs 2.5 translations of graphs Presentation 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