# Graph of a function

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.

## 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: