Math 1721.01
Worksheet #1
Include this worksheet in your "writing assignments" notebook
Graphing Shortcuts and Intuition
Using your graphing calculator, sketch the following graphs on the same coordinate axis:
a. y  x 2
b. y  x 2  4
c. y  x 2  1
Next use your calculator to draw the graphs of
a. y  cos( x)
b. y  cos( x)  2
c. y  cos( x)  5
Do you notice a pattern? Describe how the graph of y  f ( x) is related to the graph of
y  f ( x)  k and y  f ( x)  k if k is a positive number. In particular, if ( a , b) is a point
on the graph of y  f ( x) , what can you say about the point (a , b  k ) ?
Now sketch the graphs of
a. y  ( x  2) 2
b. y  ( x  4) 2
c. y  ( x  10) 2
Next sketch the graphs of
a. y  x
b. y  x  4
c. y  x  2
Describe how the graph of y  f ( x) is related to the graph of y  f ( x  k ) . In particular,
if ( a , b) is a point on the graph of y  f ( x) , what can you say about the point (a  k , b) ?
Now consider combining the two translations.
Suppose that f ( x)  x 2
Without using your graphing calculator, sketch the graph of
y  f ( x  2)  4  ( x  2) 2  4
Next suppose that f ( x)  x .
Without using your graphing calculator, sketch the graph of y  f ( x  3)  1  x  3  1.
Use this graph to determine the domain and range of the function g( x)  x  3  1 .
1
 5.
Use a similar technique to determine the domain and range of h( x) 
x2
Suppose f ( x)  x 3  4 x 2 . Sketch the graphs of
a. y  f ( x)
b. y  3  f ( x)  3( x 3  4 x 2 )
1
1
c. y   f ( x)   ( x 3  4 x 2 )
2
2
Suppose that f ( x)  sin( x) . Sketch the graphs of
a. y  f ( x)
b. y   f ( x)   sin( x)
c. y  4 f ( x)  4 sin( x)
Describe how the graph of y  f ( x) is related to the graph of y  k  f ( x) . Be certain to
mention the difference between k positive and negative. In particular, if the point
( a , b) is on the graph of y  f ( x) , what you say about the point (a , k  b) ?
Lastly, choose your own set of functions to investigate the relationship of the graph of
y  f ( x) to the graph of y  f ( kx) . Ask and then answer a similar set of questions as we
asked above.