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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Lesson 6­3
The Graph Translation Theorem
y ­ 3 = (x ­ 8)2
Dec 2­6:33 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
1.
December 16, 2014
In an equation y = f(x) what does replacing x with (x – h) do to the graph of the function?
The graph translates(slides) h units horizontally. Right if h>0 and left if h<0.
2.
In an equation y = f(x) what does replacing y with (y – k) do to the graph of the function? The graph translates(slides) k unit vertically. Up if k>0 and down if k<0.
3.
What translation does the notation T h, k indicate?
The graph translates(slides) h units horizontally k unit vertically. Both at the same time.
4.
Graph­Translation Theorem
5.
Study example 1
6.
Now try the following problem in your notes:
a.
Find an equation for the image of the graph of y = 5x2 under the translation
replace y with y­6 and replace x with Dec 2­7:02 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
7.
Corollary for translating parabolas
8.
Study example 2
9.
Now try the following problem in your notes:
a.
December 16, 2014
Graph y – 2 = ­2(x + 3)2
b.
How is this graph related to the graph
y = ­2x2?
The new graph is congruent to the graph of y = ­2x2 and shifted left 3 and up 2.
a =
Domain:
­2
vertex:
(­3, 2)
All Real #'s
Line of symmetry:
x = ­3
Range:
All Real #'s less than or equal to 2
Dec 2­7:22 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
10.
December 16, 2014
What is the following form for the equation of a parabola called?
y – k = a(x – h)2
Vertex Form for the equation of a parabola
11.
Using the equation y – k = a(x – h)2
a.
What is the vertex
(h, k)
b.
What is the line of symmetry?
Some times called the axis of symmetry
x = h
c.
What three things do you know if a > 0?
The parabola opens up.
There is a minimum and the minimum output value is equal to the y­value, k, of the vertex
The range is greater than or equal to k.
d.
What three things do you know if a < 0?
The parabola opens down.
There is a maximum and the maximum out put value is equal to the y­value, k, of the vertex
The range is less than or equal to k.
12.
Study example 3
13.
Now try the following problem in your notes:
a.
Sketch the graph of a = 1/2 a >0, opens up
h = 0, k = ­3
vertex is (0, ­3)
Domain:
All real numbers
Range:
All real numbers greater than or equal to ­3
b.
Give an equation for the axis of symmetry of the parabola.
Line of symmetry is x = 0
Dec 2­7:25 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
14.
Study example 4
15.
Now try the following problem in your notes:
a.
December 16, 2014
If parabola P has equation y = ­2x2, what is the equation for parabola Q?
a = h = k = Dec 2­7:14 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
vertex
vertex
December 16, 2014
#1
Jul 21­8:24 AM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
preimage
vertex
December 16, 2014
image
vertex
Dec 15­3:18 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
preimage
vertex
December 16, 2014
image
vertex
Dec 15­3:18 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
preimage
vertex
December 16, 2014
image
vertex
Dec 15­3:18 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Lesson 6­3: Vertex Form of an Equation
y ­ k = a ( x ­ h)2
Parabola
Characteristics
a > 0: opens up;
vertex is a minimum
Vertex
Axis of Symmetry
vertex is the ordered pair (h, k)
Translation
Th,k
" k" units
horizontally vertically
"h" units
a < 0: opens down;
vertex is a maximum
|a| small number:
wide parabola
|a| large number:
narrow parabola
axis of symmetry is the line with equation x = h
(1st coordinate of the vertex!)
Axis of Symmetry is the reflection line of the parabola
h < 0 moves left k < 0
moves
down
h > 0 moves right k > 0
moves
up
Jul 21­9:39 AM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Vertex form of an equation: y ­ k = a (x ­ h)2
Preimage Translation: Translation: Equation of Image Vertex of Axis of Mapping of Points:
Words Symbols Image Symmetry Th, k (x, y) y = x2
Translated 12 units left and 6 units up
T­12, 6
y ­ 6 = 1 (x + 12)2 (­12, 6)
y = ­7x2
Translated 6 units right and 2 units down
T6, ­2
y + 2 = ­7 (x ­ 6)2
(6, ­2)
x = 6
(2, ­28) ­­>
(8, ­30)
T­3, ­3
y + 3 = 5 (x + 3)2
(­3, ­3)
x = ­3
(­3, 45) ­­>
(­6, 42)
y +2 = ­2 (x ­ 9)2
(9, ­2)
x = 9
(2, ­8) ­­>
(11, ­10)
y = 4 (x + 1)2
Graph Below
(­1, 0)
x = ­1
y + 4 = ­3x2
Graph Below
(0, ­4)
x = 0
y ­ 5 = (x + 3)2
Graph Below
(­3, 5)
x = ­3
y = 5x2
y = ­2x2
Translated 3 units left and 3 units down
Translated 9 units right and 2 units down
T9, ­2
Translated 1 unit left y = ­3x2
Translated 4 units down
T0, ­4
Translated 3 units left and 5 units up
T­3, 5
y = 4x2
y = x2
T­1, 0
(­2, 4) ­­>
x = ­12
(­14, 10)
Jul 21­10:04 AM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Mar 4­9:42 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Mar 4­9:42 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Mar 4­9:42 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Mar 4­9:42 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
a = h = December 16, 2014
k = Dec 2­6:39 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
a = h = December 16, 2014
k = Dec 2­6:33 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
a = h = December 16, 2014
k = Dec 2­6:43 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
a = h = December 16, 2014
k = Dec 2­6:43 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
a = h = December 16, 2014
k = Dec 2­6:45 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
a = h = k = December 16, 2014
a = h = k = Dec 2­6:45 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
The solutions to the second equation are 3 large (to the right) than the solutions to the first equations
Dec 2­6:45 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:45 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:46 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:46 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:46 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:48 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:48 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:49 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:49 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Mar 4­9:47 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 2­6:52 PM
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AdvAlg6.3TheGraphTranslationTheorem.notebook
December 16, 2014
Dec 1­4:23 PM
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Attachments
bestCopy of ParabolaWS.xlsx