Difference between revisions of "Graph"
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− | Luckily the quadratic factors as <math>(2x+3)(x-1)</math> making the roots <math>x=-\frac 32</math> and <math>x=1</math>. The quadratic can only switch signs as its zeros. So picking one point less than <math>-\frac 32</math> and plugging it in will | + | Luckily the quadratic factors as <math>(2x+3)(x-1)</math> making the roots <math>x=-\frac 32</math> and <math>x=1</math>. The quadratic can only switch signs as its zeros. So picking one point less than <math>-\frac 32</math> and plugging it in will determine whether the graph is above or below the <math>x</math>-axis for all <math>x</math> on the interval <math>\left(-\infty, -\frac 32\right).</math> Since <math>f(-2)=3</math> is positive, the graph is above the <math>x</math>-axis. |
− | Likewise, | + | Likewise, a sign analysis on the intervals <math>\left(-\frac 32, 1\right)</math> and <math>(1, \infty)</math> allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline: |
<center>[[Image:Parabola1.PNG]]</center> | <center>[[Image:Parabola1.PNG]]</center> |
Revision as of 00:21, 13 August 2006
A graph is a visual representation of a function. If then the point lies on the graph of .
Contents
[hide]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: