Graph
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 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: