# Math 22 Graph of Equation

## The Graph of an Equation

The graph of an equation is the set of all points that are solutions of the equation.

In this section, we use point-plotting method. With this method, you construct a table of values that consists of several solution points of the equation

For example, sketch the graph of $y=2x+1$ . We can construct the table below by plugging points for $x$ .

 x 0 1 2 3 y=2x+1 1 3 5 7

So, we can sketch the graph from those order pairs.

## Intercepts of a Graph

Some solution points have zero as either the $x$ -coordinate or the $y$ -coordinate. These points are called intercepts because they are the points at which the graph intersects the $x$ - or $y$ -axis.

 To find $x$ -intercepts, let $y$ be zero and solve the equation for $x$ .

To find $y$ -intercepts, let $x$ be zero and solve the equation for $y$ .


Example Find the x-intercepts and y-intercepts of the function $y=x^{2}-2x$ Solution:
x-intercept: Let $y=0$ , so $x^{2}-2x=0$ , hence $x(x-2)=0$ , therefore, $x=0$ or $x=2$ y-intercept: Let $x=0$ , so $y=(0)^{2}-2(0)=0$ Answer: $(0,0)$ and $(2,0)$ are x-intercepts
$(0,0)$ is y-intercept

## Circles

 The standard form of the equation of a circle is

$(x-h)^{2}+(y-k)^{2}=r^{2}$ The point $(h,k)$ is the center of the circle, and the positive number $r$ is the radius of the circle


In general, to write an equation of a circle, we need to know radius $r$ and the center $(h,k)$ .

Example Given that the point $(2,1)$ is on the circle centered at (3,4). Find the equation of a circle.

Solution:
We need to know the radius and the center in order to write the equation. The center is given at $(3,4)$ . It is left to find the radius.
Radius is the distance between the center and a point on the circle. So, radius $r$ is the distance between $(2,1)$ and $(3,4)$ .
So, $r={\sqrt {(2-3)^{2}+(1-4)^{2}}}={\sqrt {1+9}}={\sqrt {10}}$ Now, write the equation of the circle with radius $r={\sqrt {10}}$ and center $(3,4)$ to get:
$(x-3)^{2}+(y-4)^{2}=10$ 