Math 110
In Class Exercises, Section 2.7
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As you proceed through these exercises, it may be helpful for you to fill in the following table with notes for yourself:
Vertical
Horizontal
Shift
Addition/Subtraction
Stretch/Shrink
Multiply/Divide
Outside the
Inside the
parentheses
parentheses
Graphing Functions Via Transformations
1) Outside Change Build up an intuition about what effect an 'outside change' has, by graphing the
following, using your graphing calculator. Your goal here is actually twofold: first, to figure out
what effect adding something to f(x) has on the graph, and second, to become familiar with what
the different functions look like, so you can recognize them by their graph, later on.
Parabola – parent function: f (x )  x 2
A.
1.
f (x )  x 2  2 , f (x )  x 2  4
2.
f (x )  x 2  3 , f (x )  x 2  5
What would the functions above look like if the parent function was the cubic, f (x )  x 3 ? I.e.,
B.
1.
2.
f x   x 2  2 , f x   x 2  4
f x   x 2  3 , f x   x 2  5
C.
What if the parent function is absolute value, f (x )  x ?
D.
What if the parent function is square root, f (x )  x ?
2) Sum up what you've effect (on a function's graph) is obtained from adding/subtracting a
number to/from a function, in English:
3) Given an intuitive explanation (in English) why the change to the equation produces the change
in the graph. Does the effect of the change depend on what f(x) is?
4) Inside Change Build up an intuition about what effect an 'inside change' has, by graphing the
following, using your graphing calculator:
A. Parabola – parent function: f (x )  x 2
1.
f (x )  (x  2)2 , f (x )  (x  4)2
2.
f (x )  (x  3)2 , f (x )  (x  5)2
B. What would the functions above look like if the parent function was the cubic?
C.
What if the parent function is absolute value, f (x )  x ?
D.
What if the parent function is square root, f (x )  x ?
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Math 110
In Class Exercises, Section 2.7
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5) Sum up what you've observed, in English:
6) Given an intuitive explanation (in English) why the change to the equation produces the change
in the graph. Does the effect of the change depend on what f(x) is?
7) Find the formula of the function, given the
following picture
8) Find the formula of the function, given the
following picture
9) Find a formula of the function, given the following description:
Start with y = x2.
3
units:
2
3
Write the formula for the function that will move the graph to the left by
units AND down by 2:
2
Write the formula for the function that will move the graph to the left by
10) Find a formula of the function, given the following description:
1.
Shape of f (x )  x , moved 2 up and 3 right
11) Reflection Build up an intuition about what effect multiplying by -1 has, by graphing the
following, using your graphing calculator:
3
3
A.
Across the x-axis: f x   x , g x    x
 
B.
Across the y-axis: f (x )  x , g (x )  x
C.
Both axes: f (x )  x , g (x )   x
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Math 110
In Class Exercises, Section 2.7
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12) Sum up what you've observed, in English:
13) Given an intuitive explanation (in English) why the change to the equation produces the
change in the graph. Does the effect of the change depend on what f(x) is?
14) Find the formula of the function, given the
following picture:
15) Find the formula of the function, given the
following picture:
16) Find the formula of the function, given the following picture:
17) Vertical stretch/shrink - OUTSIDE Build up an intuition about what effect an 'outside vertical
stretch/compression' has, by graphing the following, using your graphing calculator:
1.
f (x )  x 2  g (x )  2x 2 , g (x )  3x 2 , g (x )  4x 2
2.
f (x )  x 2  g (x )  0.6x 2 ,
Math 110
g (x )  52 x 2 ,
g (x )  0.2x 2
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Math 110
In Class Exercises, Section 2.7
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18) Sum up what you've observed, in English:
19) Given an intuitive explanation (in English) why the change to the equation produces the
change in the graph. Does the effect of the change depend on what f(x) is?
20) Horizontal stretch/shrink – INSIDE Build up an intuition about what effect an 'inside vertical
stretch/compression' has, by graphing the following, using your graphing calculator:
1.
2.
f (x )  sin x  g (x )  sin  2x  , g (x )  sin  4x  , g (x )  sin  8x 
f (x )  sin x  g (x )  sin 0.6x  , g (x )  sin  52 x  , g (x )  sin 0.2x 
21) Sum up what you've observed, in English:
22) Given an intuitive explanation (in English) why the change to the equation produces the
change in the graph. Does the effect of the change depend on what f(x) is?
23) Familiarize yourself with what the graph of the function sin(x) (pronounced "Sine of X")) by
graphing it in your graphing calculator. Notice the periodic (repeating) pattern that it establishes. Later (in
Math 120) you'll look at this function much more, including using it to model things like rainfall over the year (which
also varies in this periodic, repeating pattern).
You should makes sure that you're comfortable with this function by filling in the following information:
What is the point at which the function crosses the x-axis, nearest x = 0?
Looking rightwards, what is the next point at which the function crosses the x-axis, x > 0?
What is the maximum value the function attains?
What is the minimum value the function attains?
24) Find the formula of the function, given the
following picture:
Math 110
25) Find the formula of the function, given the
following picture:
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Math 110
In Class Exercises, Section 2.7
26) Find the formula of the function, given the
following picture:
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27) Find the formula of the function, given the
following picture:
28) Find a formula of the function, given the following description:
1.
Shape of f (x )  x , moved 3 down, 4 left, and reflected across the x-axis.
2.
Shape of f (x )  x , shifted 2 units right, 5 units up, reflected across the y-axis
1.
Shape of f (x )  x 4 , translated 5 units left, 3 up, reflected across the x-axis, and shrunk vertically by
a factor of 0.5.
2.
Shape of f (x )  3 x , shifted 2 units down, 1 unit right, reflected across the y-axis, and stretched
vertically by a factor of 4.
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Math 110
In Class Exercises, Section 2.7
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Given the picture of the function f(x) above, draw the graph of the function that would result by
performing the following transformations on f(x):
1
f  x  3  1
2
Draw your graph directly onto the above graph.
(For the squiggly parts: draw it as best you can, and be clear about the new locations of the graph when
x = -10, x = -5, x = -3, x = 2, x = 8, and x = +10).
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