Difference between revisions of "Graph"
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=== Solution === | === Solution === | ||
| − | + | To graph a line, it is necesasry to find two points <math>(x,y)</math> that satisfy <math>2x + 3y = 24</math>. Letting <math>x=0</math> gives <math>3y = 24\Leftrightarrow y = 8</math>. So <math>(0,8)</math> is one point on the graph. | |
| − | + | Find another point by letting <math>y=0</math>. Plugging this in and solving gives <math>x=12</math>. So <math>(12,0)</math> is our other point. | |
| − | Now | + | Now plot these in the coordinate plane and draw a line through them: |
<center>[[Image:Twopoints2.PNG]]</center> | <center>[[Image:Twopoints2.PNG]]</center> | ||
| − | The arrowheads on the ends of the line segment indicate that the line goes on [[infinite]]ly in both directions. | + | The arrowheads on the ends of the line segment indicate that the line goes on [[infinite]]ly in both directions. |
== Graphing Polynomials == | == Graphing Polynomials == | ||
Revision as of 00:14, 13 August 2006
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.
.PNG)
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 tell us whether the graph is above or below the -asis for all on the interval 
Since 
is positive, the graph is above the -axis.
Likewise, we do a sign analysis on the intervals 
and 
, draw a smooth curve curve through the zeros using this information as a guideline:
Problem
Graph 
.
Solution
First, we find the zeros of the function. Plugging in numbers, we find works. After synthetic division we have 
. Plugging numbers into the cubic, we find that 
works. Thus, we have 
. Finally, we can factor the quadratic giving us 
. So we have roots at 1 and -3 and a double root at 2. Our final graph looks like:
