Graph of a function
From AoPSWiki
A graph is a visual representation of a function. If 
then the point 
lies on the graph of 
.
Contents |
Graphing Points
A single point is the simplest thing to graph. The graph of 
would be a dot 2 units to the right of 
-axis and 5 units above the 
-axis.
Graphing Lines
Given two distinct points on a line, one can construct the whole line. So one way to graph a line given its equation is to just find two points on it and to draw a straight line through them.
Problem
Graph the line 
.
Solution
To graph a line, it is necesasry to find two points that satisfy . Letting 
gives 
. So 
is one point on the graph.
Find another point by letting 
. Plugging this in and solving gives 
. So 
is our other point.
Now plot these in the coordinate plane and draw a line through them:
The arrowheads on the ends of the line segment indicate that the line goes on infinitely in both directions.
Graphing Polynomials
The first step in graphing a polynomial, 
, is to find the zeros of 
. Then a smooth curve should be drawn through the zeros accounting for multiple roots and making sure the signs match up (i.e. the graph is above the -axis when the polynomial is positive and below it when the polynomial is negative). This process is best understood through examples.
Problem
Graph the parabola 
.
Solution
Luckily the quadratic factors as 
making the roots 
and 
. The quadratic can only switch signs as its zeros. So picking one point less than 
and plugging it in will determine whether the graph is above or below the -axis for all on the interval 
Since 
is positive, the graph is above the -axis.
Likewise, a sign analysis on the intervals 
and 
allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline:
Problem
Graph 
.
Solution
First, find the zeros of the function. Note that if or 
, . After synthetic division, the polynomial reduces to 
. Factor the quadratic gives 
. So the roots are 1 and -3 and a double root at 2. The final graph looks like:
