CPhill gave you lots of great ideas.
If you want to plot $$y=x^2+5x+4$$ without plotting a whole lot of points this is the best way.
First the y intercept is where x=0 so substitute in x=0 and you will get y=4 (the number tacked on the end- this is called the constant)
This in not an overly important point to find but it is very easy to find (0,4)
Now lets look for some important points. If they exist we want to find the x intercepts these can also be called the roots or less often the zeros of the equation. You can factorise and find them that way or you can find them using the quadratic equation.
I'm going to factorise. I need 2 numbers that multiply to 4 and add to 5. That is easy, 4 and 1
$$y=(x+4)(x+1)$$ I want the answers when y=0 (That is the equation of the x axis)
$$0=(x+4)(x+1)$$ If 2 things multiply to give zero then one of them must be zero.
x+4=0 or x+1=0
x=-4 or x=-1
Thes are the roots so plot them.
now parabolas are symetrical so the axis of symmetry must be
$$x=\frac{-4+-1}{2}\\
x=\:-2.5$$
Find the vertex by sub'ing in x=-2.5 to find the y value
Then you have the vertex and 2 roots and that is enough to fully define your parabola.
+118725 
