Shifting and Reflecting

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Shifting and Reflecting

Six Basic Functions

Below are six basic functions:
1. $Graph of the absolute value of x. V shaped$
2. $Graph of y = x squared, parabola.$
3. $Graph of y = x cubed. Down on the left, up on the right and pretty flat through the origin.$
4. $Graph of y = 1/x. vertical asymptote at the origin and horizontal asymptote along the x-axis.$
5. $Graph of y = the square root of x. Starts at the orgin, curves up to (1,1), keeps going up as x increases, but not very steep for larger x values..$
6. $Graph of y = x to the 1/3. Goes through the origin, increases from left to right, but not very steep when it is far from the origin and it is very steep near the origin.$
  
  Memorize the shapes of these functions.

Horizontal Shifting

Consider the graphs

y =

(x + 0)²
(x + 1)²
(x + 2)²
(x + 3)²

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

Vertical Shifting

Consider the graphs

y = $Graphs of the 4 cubics y = x^3 + a with a = 0, 1, 2, and 3, shifted up 1 as the a increases by 1.$

x³
x³ + 1
x³ + 2
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.

Reflecting About the x-axis

Consider the graphs

y = x² and y = -x²

$Graphs of the 2 parabolas y = x^2 and y = -x^2. The first is concave up and the second is concave down.$

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

Reflecting About the y-axis.

Exercise:

Use the calculator to graph

$ y = \sqrt{x}$

and

$ y = -\sqrt{x}$

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

Stretching and Compressing

Exercises

Graph 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

y = x² - 10

$ y = \sqrt{x}$

y = -|x - 5| + 3

For an interactive investigation of the shifting rules go to

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

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
$Graph of a function that, far left goes down, "Dec", then goes up, "Inc", then on the far right goes down again, "Dec".$
        x < y

then

        f(x) > f(y)

Example:

The curve

        y = x²

is increasing on (0, $\infty$ ) and decreasing on ($-\infty$ ,0)

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