Difference between revisions of "Graph"
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=== Problem === | === Problem === | ||
| − | Graph <math>x^4 - 2x^3 -7x^2 +20x -12</math>. | + | Graph <math>y = x^4 - 2x^3 -7x^2 +20x -12</math>. |
=== Solution === | === Solution === | ||
| − | First, | + | First, find the zeros of the function. Note that if <math>x=1</math> or <math>x=2</math>, <math>y=0</math>. After [[synthetic division]], the polynomial reduces to <math>y=(x-1)(x-2)(x^2+x-6)</math>. Factor the quadratic gives <math>(x-1)(x-2)^2(x+3)</math>. So the roots are 1 and -3 and a double root at 2. The final graph looks like: |
<center>[[Image:Quartic1.PNG]]</center> | <center>[[Image:Quartic1.PNG]]</center> | ||
Revision as of 00:17, 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, 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:
