Exploring Transformations of Graphs using your TI84+
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f ( x)  f ( x)  b
1. Set the window : Xmin= -10, Xmax=10, Xscl=1, Ymin= -10, Ymax=10, Yscl=1, Xres=1 (pressing
ZOOM then 6:ZStandard does this automatically)
2. On the same screen draw these three graphs: y  x 2
(here
y  x2  5 y  x2  7
f ( x)  x 2 )
Note: in Y=, highlighting the slants before Y1, Y2 or Y3 and pressing enter gives you different
options for the line style of your graphs – you can use these to make it easier to distinguish
between the different curves on the screen – bold, dotted and normal, for example.
Sketch the graphs on the same set of axes (label each graph with its equation):
3. Predict where the curve y  x 2  8 will be. Use your calculator to check your prediction.
4. Clear the screen and then draw these three graphs on the same screen:
y  ( x  2) 2
y  ( x  2) 2  3
y  ( x  2) 2  6
(here f ( x)  ( x  2) 2 ). Sketch the graphs on the same set of axes (label each graph with its
equation):
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In general, how is the graph of y  f ( x)  b (where b is any real number) obtained from the
graph of y  f (x) ?
Try to explain why this is so
f ( x)  f ( x  a )
1. Press MODE and check that your calculator is set to degrees. Now set the window:
Xmin= -360, Xmax= 360, Xscl=10, Ymin= - 1.5, Ymax= 1.5, Yscl=1, Xres=1
2. On the same screen draw these three graphs:
y  cos( x)
y  cos( x  90)
y  cos( x  45)
(here f ( x)  cos( x) )
Sketch the graphs on the same set of axes (label each graph with its equation):
3. Clear the screen. Then draw the graph of y  cos(x) again. Predict where the graph of
y  cos( x  60) will be. Use your calculator to check your prediction.
4. Set the window : Xmin= -10, Xmax=10, Xscl=1, Ymin= -10, Ymax=10, Yscl=1, Xres=1. Clear the
screen and then draw these three graphs on the same screen: y  x 2 y  ( x  5) 2 y  ( x  2) 2 .
(Here f ( x)  x 2 )
Sketch the graphs on the same set of axes (label each graph with its equation):
Predict where the graph of y  ( x  6) 2 will be. Use your calculator to check your prediction.
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5. Clear the screen and then draw these three graphs on the same screen:
(Here f ( x)  2 x )
y  2x
y  2 x 6 y  2 x  7
Sketch the graphs on the same set of axes (label each graph with its equation):
In general, how is the graph of y  f ( x  a) (where a is any real number) obtained from the
graph of y  f (x) ?
Try to explain why this is so
f ( x)  pf ( x)
1. Set the window: Xmin= -360, Xmax=360, Xscl=90, Ymin= - 2.5, Ymax=2.5, Yscl=1, Xres=1. On
the same screen draw these three graphs:
y  sin( x) y  2 sin( x)
(here f ( x)  sin( x) )
y  0.6 sin( x)
Sketch the graphs on the same set of axes (label each graph with its equation):
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2. Set the window: Xmin= - 5, Xmax= 5, Xscl=1, Ymin= - 20, Ymax=25, Yscl=1, Xres=1. On the
same screen draw these three graphs:
y  0.5( x 3  x 2  6 x)
y  x 3  x 2  6x
y  3( x 3  x 2  6 x)
(here f ( x)  x 3  x 2  6 x )
Remember you can use MATH 3 to get the cubic function.
Sketch the graphs on the same set of axes (label each graph with its equation):
In general, how is the graph of y  pf (x) (where p is any real number) obtained from the graph
of y  f (x) ?
Try to explain why this is so
 x
f ( x)  f  
q
1. Set the window: Xmin= -360, Xmax=360, Xscl=90, Ymin= - 1.5, Ymax= 1.5, Yscl=1, Xres=1. On
the same screen draw these three graphs: y  sin( x)
(here f ( x)  sin x )
 x
y  sin   y  sin( 2 x)
2
Remember: in Y=, highlighting the slants before Y1, Y2 or Y3 and pressing enter gives you different options
for the line style of your graphs – you can use these to make it easier to distinguish between the different
curves on the screen. Sketch the graphs on the same set of axes (label each graph with its equation):
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 x
4
Predict where the graph of y  sin   will be. Use your calculator to check your prediction.
2. Suppose f ( x)  x 3  2 x 2  24 x . Set the window to: Xmin= -20, Xmax=20, Xscl=1, Ymin= -100,
Ymax=100, Yscl=10, Xres=1. On the same screen draw the graphs of
 x
y  f 
2
y  f ( x)
 x
y  f   . Sketch and label them here:
3
Important GDC tip:
 x
2
In question (2) above you could have found f  
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algebraically first:
2
3
x2
 x  x
 x
 x x
f       2   24  

 12 x and typed this in as Y2. A similar approach could be
2
2 2
2
2 8
used for Y3. This is unnecessarily long. With the GDC you can achieve the same three required graphs
like this:
or even better like this:
To obtain Y1(X/2) this is what you do: VARS select Y-VARS select 1:FUNCTION select 1:Y1
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3. Set the window: Xmin= -7, Xmax=7, Xscl=1, Ymin= - 1, Ymax= 10, Yscl=1, Xres=1. On the same
screen draw these three graphs: y  3x 2
y  x2 y 
Sketch and label them here:
x2
2
y
x2
4
(here f ( x)  x 2 )
 x
In general, how is the graph of y  f   (where q is any real number) obtained from the graph
q
of y  f (x) ?
Try to explain why this is so
f ( x)  f ( x)
Set the window : Xmin= -10, Xmax=10, Xscl=1, Ymin= -10, Ymax=10, Yscl=1, Xres=1 (pressing
ZOOM then 6:ZStandard does this automatically)
1. Suppose f(x) = (x-2)2 What is f(-x)?
Use your calculator to draw the graphs of y = f(x) and y = f(-x) on the same screen. Sketch the
two graphs on the same set of axes below (label each graph with its equation):
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2. Suppose f(x) = 3x+7
What is f(-x)?
Use your calculator to draw the graphs of y = f(x) and y = f(-x) on the same screen.
Sketch the two graphs on the same set of axes below (label each graph with its equation):
In general, how is the graph of y  f (  x) obtained from the graph of y  f (x) ?
Try to explain why this is so
f ( x)   f ( x)
1. Suppose f(x) = x2
What is -f(x)?
Use your calculator to draw the graphs of y = f(x) and y = -f(x) on the same screen.
Sketch the two graphs on the same set of axes below (label each graph with its equation):
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2. Suppose f(x) = x  3
What is -f(x)?
Use your calculator to draw the graphs of y = f(x) and y = -f(x) on the same screen.
Sketch the two graphs on the same set of axes below (label each graph with its equation):
In general, how is the graph of y   f (x) obtained from the graph of y  f (x) ?
Try to explain why this is so
f ( x)  f 1 ( x)
Set the window by pressing ZOOM then 5:ZSquare to get the same scale on both axes
1. Suppose f(x) = 2x - 5
What is f-1(x)?
Use your calculator to draw the graphs of y = f(x), y = x and y = f-1(x) on the same screen.
Sketch the three graphs on the same set of axes below (label each graph with its equation):
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2. Suppose f(x) = x2.
What is f-1(x)?
Use your calculator to draw the graphs of y = f(x), y = x and y = f-1(x) on the same screen.
Sketch the three graphs on the same set of axes below (label each graph with its equation):
In general, how is the graph of y  f
1
( x) obtained from the graph of y  f (x) ?
Try to explain why this is so
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