Shifting and Reflecting

I.  Homework

II.  Five Basic Functions

We will learn five basic functions:

A)  y = |x|

B)  y = x²

C)  y = x³

D)  y = 1/x

E)  y = sqrt(x)

We will graph these and do calisthenics.

III.  Horizontal Shifting

 We will demo with a the TI85 the following:

A)  y = 1/[x - {0,1,2,3}]

B)  y = (x + {0,1,2,3})²  

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

IV.  Vertical Shifting

We will demo with a the TI85 the following:

A)  y = |x| + {0,1,2,3}

B)  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.

V.  Reflecting About the x-axis

We will demo with a the TI85 the following:

 y = x²  and y = -x²  

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

VI.  Reflecting About the y-axis.

We will demo with a the TI85 the following:

y = sqrt(x) and y = sqrt(-x)

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

VII.  Stretching and Compressing

We will demo with a the TI85 the following:

y = {1,2,3,4}x³

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

A)  y = x² - 10

B)  y = sqrt(x - 2)

C)  y = -|x - 5| + 3 

VIII.  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,infinity) and decreasing on (-infinity,0)