Transformations of graphs
General objectives:
To explore geometrical transformations and their
effects on the graphs of functions
To relate transformations with the variations in the
equation of a function.
You must be able to recognise the graphs of
different functions and to draw a sketch
without calculator.
constant function
identity function
absolute value function
quadratic function
square root function
cubic function
reciprocal funtion
exponential function
trigonometric funtion
inverse square function
Lesson 1
Transformations of graphs
TRANSLATIONS
To explore vertical and horizontal translations and
their effects on the graphs of functions
To relate translations with the variations in the
equation of a function.
Based on the graph of f(x) , draw the graph of f(x)+3
What will the graph of f(x) ­ 2 be ?
Vertical translation.ggb
Based on the graph of f(x) , draw the graph of f(x) ­ 1
y=f(x)
This is the graph of
f(x) ­ 4
y = f(x) .Sketch the graph of
y=f(x)
Conclusions:
y = f (x ) + c
If y = f (x)
translates vertically the graph of
y = f (x), c units.
translation vector:
• If
c>0
it moves upwards.
• If
c<0
it moves downwards.
If
, find f (x ­ 2)
Based on the graph of
f(x­2)=
, draw the graph of y = f (x ­ 2)
y=f(x)
Horizontal translation.ggb
Based on the graph of
f(x­2)=
, draw the graph of y=f(x­2)
Horizontal translation.ggb
y=f(x)
Use your GDC to draw the graph of
and
y=f(x)
Compare
and
y=f(x)
What can you predict about the graph of g?
asymptotes?
Conclusions:
If y= f( x )
b >0
y= f( x ­ b)
translates horizontally the graph of
y= f( x ),
b units to the right.
translation vector:
y= f( x + b)
translates horizontally the graph of
y= f( x ),
b units to the left.
translation vector:
Exercise 1: Sketch
the graph of: f(x) ­ 2 ; f (x ­ 3)
f(x)
Exercise 2: Sketch
the graph of the function
Hence, sketch the graph of
Exercise 3: • Sketch the graph of
• Sketch the graph of
• Find g(x) in its simplest form.
Exercise 4:
• Sketch the graph of
• Sketch the graph of
and indicate clearly any asymptote.
and indicate clearly any asymptote.
Exercise 5:
• Sketch the graph of
• Sketch the graph of
and indicate clearly any asymptote.
and indicate clearly any asymptote.
is translated and its
The function
vertex is now the point (­2, ­3).
What translations were applied to function f ?
Will the order in which you apply the translations
affect the final result?
Give the formula of the new function.
Lesson 2
Given
same set of axes.
sketch the graphs of the following functions on the
Lesson 2
Transformations of graphs
STRETCH
REFLECTIONS
To explore stretches and reflections and their
effects on the graphs of functions
To relate stretches and reflections with the
variations in the equation of a function.
Set up your GDC in "degrees".
Prepare domain :
With your calculator , plot the graphs of
What is the effect on the graph of
will produce if y= a f(x) ?
y= f(x) that "a"
Vertical stretch.ggb
y = sin x
Verify your conclusion for the graphs of
Conclusions:
If
y= f( x )
a >1
y= a f( x )
stretches vertically the graph of
y= f( x ),
1
y= a f( x )
scale factor:
a
stretches vertically the graph of
y= f( x ),
scale factor: 1
a
Use your calculator to draw the graph of
, for
On the same grid draw
What effect produces the "2" of f(2x) on the graph of f(x)?
Horizontal stretch.ggb
y= sin x
y= sin (2x)
horizontal stretch , scale factor
Now , draw
and
y= sin (2x)
y= sin x
y= sin (½x)
y= sin (½x)
horizontal stretch , scale factor 2
Conclusions:
If
y= f( x )
a >1
y= f( ax )
stretches horizontally the graph of
y= f( x ),
y= f( 1a x )
scale factor:
1
a
stretches horizontally the graph of
y= f( x ),
scale factor:
a
A point (x, y) on the graph of y= f(x)is
transformed to the point
y=f(ax)
in the graph of
Exercise 1:
• The function f is defined by
• Sketch the function with the help of your GDC.
• Describe the geometric transformation that
will apply to the graph of f.
Use your calculator to draw the graphs of
and
What geometrical transformation does -f(x) represent?
Based on the graph of y= f(x) , draw the graph of y = ­ f(x).
Write down the equations of both lines .
Reflect the following function about the y-axis.
y =g(x)
y = f(x)
Complete:
g(2) = f (.....)
g(­2) = f (.....)
g(1) = f (.....)
g(­1) = f (.....)
g(x) = f (.......)
Conclusions:
y = ­f( x )
If y= f( x )
reflects the graph of
about
y= f(­ x )
y= f( x ),
the x-axis.
reflects the graph of
about the y-axis.
y= f( x ),
Match graphs and formulae:
f(x)
f(x)+3
2f(x)
f(x­2)
f(x)
b f(x)+c
f(x)+a
f(x+a)
b f(x)
y = a f(x)
y = f(ax)
y = f(x)
vertical translation
Translations
c
horizontal translation
vertical stretch (a)
Stretchs
horizontal stretch (
reflection x-axis
Reflections
reflection y-axis
ALibraryOfFunctionsWithTransformations.nbp
)
To revise this topic at home:
http://enlvm.usu.edu/ma/nav/activity.jsp?sid=__shared&cid=emready@trfns&lid=136
When you feel ready, self-assessment:
http://archives.math.utk.edu/visual.calculus/0/shifting.7/index.html
Solve Book page 86
Exercise 3A
Exercises
1) c) d)
2) c) d)
3)b)
4)c)d)
5)c)d)
8) 9) 10) 11) 12) 13)
Attachments
Vertical translation exponential.ggb
Horizontal translation quadratic.ggb
ALibraryOfFunctionsWithTransformations.nbp
Horizontal stretch.ggb
Vertical translation.ggb
Horizontal translation.ggb
Vertical stretch.ggb