Shifting and Reflecting

1. Six Basic Functions

Below are six basic functions:

1. Memorize the shapes of these functions.

2. Horizontal Shifting

Consider the graphs

y =

1. (x + 0)2

2. (x + 1)2

3. (x + 2)2

4. (x + 3)3

Exercise

Use the list features of a calculator to sketch the graph of

y = 1/[x - {0,1,2,3}]

 Rule1:  f(x - a) = f(x) shifted a units to the right. Rule 2:  f(x + a) = f(x) shifted a units to the left

3. Vertical Shifting

Consider the graphs

y =

1. x3

2. x3 + 1

3. x3 + 2

4. x3 + 3

Exercise

Use the list features of a calculator to sketch the graph of

y = x3  - {0,1,2,3}

 Rule 3:  f(x ) + a = f(x) shifted a units up. Rule 4:  f(x) - a = f(x) shifted a units down.

Consider the graphs

y = x2  and y = -x2

 Rule 5:  -f(x ) = f(x)  reflected about the x-axis.

Exercise:

Use the calculator to graph

and

 Rule 6:  f(-x ) = f(x) reflected about the y-axis.

6. Stretching and Compressing

Exercises

Graph the following

1. y = {1,2,3,4}x3

2. y = {1/2,1/3,1/4,1/5}x3

 Rule 7:  cf(x ) = f(x) (for c > 1) stretched vertically. Rule 8:  cf(x ) = f(x) (c < 1) compressed vertically.

We will do some examples (including the graph of the winnings for the gambler and for the casino.

Exercises:  Graph the following

1.  y = x2 - 10

2. y = sqrt(x - 2)

3.  y = -|x - 5| + 3

For an interactive investigation of the shifting rules go to

http://mathcsjava.emporia.edu/~greenlar/Shifter/Shifter1.html

1. Increasing and Decreasing Functions

A function is called increasing if as an object moves from left to right, it is moving upwards along the graph.

If

x < y

then

f(x) < f(y)

A function is called decreasing if as an object moves from left to right, it is moving downwards along the graph.

If

x <  y

then

f(x) > f(y)

Example:

The curve

y = x2

is increasing on (0, ) and decreasing on (-,0)

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