# Thread: What steps must I take to graph equations such as y=y/2+1

1. ## What steps must I take to graph equations such as y=y/2+1

So I was wondering how do I solve equations where for example it is in the form y=y/2+x where instead of being in the standard form y=mx+b we have something where y = my + nx + b (where in the example I gave m = 1/2 n = 1 and b = 0). What steps should I take to figure out how to graph this? According the graphing application I am using (Grapher) the slope appears to be 2 but, how do I figure this stuff out?

Also, to further this what if I wanted to graph the equation:
y=ay^2 + ax^2 + my + nx + b?

What should I look into to do this?

2. Originally Posted by thyrgle
So I was wondering how do I solve equations where for example it is in the form y=y/2+x where instead of being in the standard form y=mx+b we have something where y = my + nx + b (where in the example I gave m = 1/2 n = 1 and b = 0). What steps should I take to figure out how to graph this? According the graphing application I am using (Grapher) the slope appears to be 2 but, how do I figure this stuff out?
Solve the equation
$y=\frac12 y + x$ for y:

$y=\frac12 y + x~\implies~\frac12 y = x~\implies~y = 2x$

Also, to further this what if I wanted to graph the equation:
y=ay^2 + ax^2 + my + nx + b?

What should I look into to do this?
The graph of the 2nd equation is a circle. To draw a circle you need to know the coordinates of the center and the length of the radius:

$y=ay^2 + ax^2 + my + nx + b$

$0=ay^2 + ax^2 + nx + my - y + b$

$0=y^2 + x^2 + \frac na x+ \frac{m-1}{a}y + \frac ba$

Now complete the squares:

$y^2 + \frac{m-1}{a}y +\left(\frac{m-1}{2a} \right)^2 + x^2 + \frac na x +\left(\frac{n}{2a} \right)^2= - \frac ba+\left(\frac{m-1}{2a} \right)^2 +\left(\frac{n}{2a} \right)^2$

$\left(y +\left(\frac{m-1}{2a}\right) \right)^2 + \left(x +\left(\frac{n}{2a} \right) \right)^2= - \frac ba+\left(\frac{m-1}{2a} \right)^2 +\left(\frac{n}{2a} \right)^2$

So the center is at $M\left(\left(-\frac{n}{2a} \right)\ ,\ \left(-\frac{m-1}{2a}\right) \right)$ and $r^2= - \frac ba+\left(\frac{m-1}{2a} \right)^2 +\left(\frac{n}{2a} \right)^2$