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1.2A Stretches. The graph of y + 3 = f(x) is the graph of f ( x ) translated … up 3 units left 3 units down 3 units right 3 units. x. 2. The graph of f(x) + 4 is the graph of f ( x ) translated … - PowerPoint PPT Presentation
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Math 30-1 1
1.2A Stretches1. The graph of y + 3 = f(x) is the graph of f(x) translated…
up 3 units left 3 units down 3 units right 3 units
2. The graph of f(x) + 4 is the graph of f(x) translated…
4 units up 4 units left 4 units down 4 units right
3. The graph of f(x – 7) + 6 is the graph of f(x) that has been translated..
7 units left, 6 units up 7 units left, 6 units down
7 units right, 6 units down 7 units right, 6 units up
4. In general, the graph of f(x – h) + k, where h and k are positive, as compared to the parent function graph f(x), is translated
h units left and k units up h units right and k units up h units left and k units down h units right and k units down
x
x
x
x
Math 30-1 2
5. When the output of a function y = f(x) is multiplied by -1, the result, y = -f(x), is a reflection of the graph in the a. x – axisb. y – axisc. line y = x.
6. When y = f(x) is transformed to y = f(-x), then (x, y) is transformed to d. (-x, -y)e. (x , -y)f. (-x, y)
Math 30-1 3
Function Transformations: Vertical Stretch
1.2A Vertical Stretches
Vertical Stretches: The effect of parameter a. ( )y af xA stretch changes the shape of the graph. (Translations changed the position of a graph.)
( )y af x|a| describes a vertical stretch about the x-axis.
An Invariant point is a point on a graph that remains unchanged after a transformation.
Math 30-1 4
In general, for any function y = f(x), the graph of the function y =a f(x) has been vertically stretched about the x-axis by a factor of |a| .
The point (x, y) → (x, ay). Only the y coordinates are affected.
Vertical Stretches
Invariant points are on the line of stretch, the x-axis. are the x-intercepts.
When |a| > 1, the points on the graph move farther away from the x-axis.
When |a|< 1, the points on the graph move closer to the x-axis.
( )y k fa x h
3 ( )y f x Vertical stretch by a factor of 3
1( )
3y f x Vertical stretch by a factor of ⅓
Math 30-1 5
Vertical Stretching y = af(x), |a| > 1y = f(x)A vertical stretch
about the x-axis by a factor of 2.
y = 2f(x)
Key Points
(x, y) → (x, 2y)
(-2, 0) → (-2, 0)
(-1, -7) → (-1, -14)
(1/2, 1) → (1/2, 2)
Invariant Points
(-2, 0)
(0, 0)
(1, 0)
(2, 0)
Domain and Range
Math 30-1 6
Consider
Write the equation of the function after a vertical stretch about the x-axis by a factor of ½.
Write the coordinates of the image of the point (-7, 7)
Write the transformation in function notation.
The point (x, y) maps to
y 1
2f (x)
(-7, 3.5)
List any invariant points.
How are the domain and range affected?
Vertical Stretches about the x-axis y = af(x), |a| < 1y x
1
2y x
1,2
x y
0,0
2y x
Math 30-1 7
The graph of g(x) is a transformation of f(x).
Is the transformation a translation?
Is the transformation a vertical stretch?
Invariant Point
(2, 0)→ (1, 0)
On the y-axis
(x, y)→(½x, y)
Horizontal stretchBy a factor of ½
(2 )y f x
Math 30-1 8
In general, for any function y = f(x), the graph of the function
y = f(bx) has been horizontally stretched by a factor of .
The point (x, y) →
Horizontal Stretches
1
b
x
b, y
Invariant points are on the line of stretch, the y-axis. are the y-intercepts.
When |b| > 1, the points on the graph move closer to the y-axis.
When |b|< 1, the points on the graph move farther away from the x-axis.
( )y bk af x h
(3 )y f x Horizontal stretch by a factor of ⅓
1
4y f x
Horizontal stretch by a factor of 4
Only the x coordinates are affected.
Math 30-1 9
Consider y xWrite the equation of the function after a horizontal stretch about the y-axis by a factor of 3.
y 1
3x
Write the coordinates of the image of the point (-3, 3 )
Write the transformation in function notation. y f1
3x
Write the coordinates of the image of the point (x, y) 3x, y
List any invariant points.
How are the domain and range affected?
Characteristics of Horizontal Stretches about the y-axis
3y x
Can it be written in any other way?
→ (-9, 3)
Math 30-1 10
Is This a Horizontal or Vertical Stretch of y = f(x)?
y = f(x)
The graph y = f(x) is stretched vertically about the x-axis bya factor of .
1
2
2 ( )y f x
1( )
2y f x
Math 30-1 11
Has the graph been stretched Horizontally or Vertically?
(1, 1)
2(2 )y x
24y x
Math 30-1 12
Zeros of a Function
1. What are the zeros of the function?
2. Use transformations to determine the zeros of the following functions.
b) 2 ( )y P x
1c)
2y P x
a) ( 1)y P x x = -1, 1, 4
x = -2, 0, 3
x = -2, 0, 3
x = -4, 0, 6
Math 30-1 13
Describe the transformation in words compared to the graph of a function y = f(x).
a) y = f(3x) b) 3y = f(x)
c) y = f( x) d) y = 2f(x)
Stretched horizontally by a factor of 2 about the y-axis.
Describing the Horizontal or Vertical Stretch of a Function
Stretched horizontally by a factor of about the y-axis.
1
3
Stretch vertically by a factor
of about the x-axis.1
3
1
2Stretched vertically by a factor
of about the x-axis.1
2
Math 30-1 14
Consider the graph of a function y = f(x)
The transformation described by y = f(2x+4) is horizontal stretch about the y-axis by a factor of ½.
The translation described by y = f(2x + 4) is horizontal shift of 4 units to the left.
Translations must be factored ( ( ))y k af b x h
y = f(2(x + 2))
Math 30-1 15
The graph of the function y = f(x) is transformed as described. Write the new equation in the form y = af(bx).
a) Stretched horizontal by a factor of one-third about the y-axis, and stretched vertically about the x-axis by a factor of two.
y = 2f(3x)
b) Stretched horizontally by a factor of two about the y-axis and translated four units to the left.
Stating the Equation of y = af(kx)
1
2y f x
14
2y f x
12
2y f x
Math 30-1 16
AssignmentPage 282, 5a,b, 6, 7a,c, 8, 13, 14c, d