# Graph of a function

A graph is a visual representation of a function. If $f(x) = y$ then the point $(x,y)$ lies on the graph of $f$.

## Graphing Points

A single point is the simplest thing to graph. The graph of $(2,5)$ would be a dot 2 units to the right of $y$-axis and 5 units above the $x$-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 $2x + 3y = 24$.

### Solution

To graph a line, it is necesasry to find two points $(x,y)$ that satisfy $2x + 3y = 24$. Letting $x=0$ gives $3y = 24\Leftrightarrow y = 8$. So $(0,8)$ is one point on the graph.

Find another point by letting $y=0$. Plugging this in and solving gives $x=12$. So $(12,0)$ 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, $p(x)$, is to find the zeros of $p(x)=0$. 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 $x$-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 $y = 2x^{2} + x - 3$.

### Solution

Luckily the quadratic factors as $(2x+3)(x-1)$ making the roots $x=-\frac 32$ and $x=1$. The quadratic can only switch signs as its zeros. So picking one point less than $-\frac 32$ and plugging it in will determine whether the graph is above or below the $x$-axis for all $x$ on the interval $\left(-\infty, -\frac 32\right).$ Since $f(-2)=3$ is positive, the graph is above the $x$-axis.

Likewise, a sign analysis on the intervals $\left(-\frac 32, 1\right)$ and $(1, \infty)$ allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline:

### Problem

Graph $y = x^4 - 2x^3 -7x^2 +20x -12$.

### Solution

First, find the zeros of the function. Note that if $x=1$ or $x=2$, $y=0$. After synthetic division, the polynomial reduces to $y=(x-1)(x-2)(x^2+x-6)$. Factor the quadratic gives $(x-1)(x-2)^2(x+3)$. So the roots are 1 and -3 and a double root at 2. The final graph looks like: