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Here are some of the most basic graphs, and the effects of making changes to the basic definitions of the functions:

Line: y=x	Parabola: y=x²	Cubic: y=x³

Change sign of coefficient: y = -x	Change sign of coefficient: y = -x²	Change sign of coefficient: y = -x³

Change to Slope: y=2x	Change to Horizontal Component: y = (x + 1)²	Change to Horizontal Component: y = (x-2)³


Change to vertical component: y = x + 3	Change to Vertical Component: y = x² - 4	Change to Vertical Component: Y = x³ + 4

OTHER BASIC GRAPHS

y = \|x\|	y = e^x	y = 1/x

NOW FOR PART 2.................

Problem: Graph the following system:
            2x + y >= 2
            4x + 3y <= 12
            (1/2) <= x <= 2
            y >= 0
  
  Solution: See the figure below.


 Problem: Graph y < x.

  Solution: First graph the equation y = x.
            However, the line must be drawn dashed
            because the less than sign tells us the
            line is not included in the 
            solution.
            Next, test a point that is located above
            the line and one that is below the line.
            Any point you pick above the line, such 
            as (0, 2), y is greater than x,
            so points above the line are not in-
            cluded in the solution.  Points below the 
            line, such as (3, -3) have a y 
            value that is less than the x
            value, so all points below the line are

            included in the solution.

SKETCHING THE GRAPH OF RATIONAL FUNCTIONS............!!!!!!!!!!!

Definition

A rational function f has the form

where g (x) and h (x) are polynomial functions.

The domain of f is the set of all real numbers except the values of x that make the denominator h (x) zero.

In what follows, we assume that g (x) and h (x) have no common factors.

Vertical Asymptotes

Let

The domain of f is the set of all real numbers except 3, since 3 makes the denominator zero and the division by zero is not allowed in mathematics. However we can try to find out how does the graph of f behave close to 3.

let us evaluate function f at values of x close to 3 such that x < 3. The values are shown in the table below:

x	1	2	2.5	2.8	2.9	2.99	2.999	2.99999
f (x)	-1	-2	-4	-10	-20	-200	-2000	-2*10⁵

Let us now evaluate f at values of x close to 3 such that x > 3.

x	5	4	3.5	3.2	3.1	3.01	3.001	3.00001
f (x)	1	2	4	10	20	200	2000	2*10⁵

The graph of f is shown below.

Notes

1 - As x approaches 3 from the left or by values smaller than 3, f (x) decreases without bound.

2 - As x approaches 3 from the right or by values larger than 3, f (x) increases without bound.

We say that the line x = 3, broken line, is the vertical asymptote for the graph of f.

In general, the line x = a is a vertical asymptote for the graph of f if f (x) either increases or decreases without bound as x approaches a from the right or from the left. This is symbolically written as:

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