View source for Graph of a function

A '''graph''' is a visual representation of a [[function]].  If <math>f(x) = y</math> then the [[point]] <math>(x,y)</math> lies on the graph of <math>f</math>.

== Graphing Points ==
A single point is the simplest thing to graph.  The graph of <math>(2,5)</math> would be a dot 2 units to the right of <math>y</math>-axis and 5 units above the <math>x</math>-axis.

<center>[[Image:Point(2,5).PNG]]</center>

== 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 <math>2x + 3y = 24</math>.

=== 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 plot these in the coordinate plane and draw a line through them:

<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.

== Graphing Polynomials ==
The first step in graphing a [[polynomial]], <math>p(x)</math>, is to find the zeros of <math>p(x)=0</math>.  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 <math>x</math>-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 <math> y = 2x^{2} + x - 3 </math>.

=== Solution ===
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, 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>

=== Problem ===
Graph <math>y = x^4 - 2x^3 -7x^2 +20x -12</math>.

=== Solution ===
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>

== Sea also ==
* [[Algebra]]