I normally use a graphing calculator, but if you need to graph by hand I normally make a table with the x-values of -5 to 5 and plug in those range of x-values to graph the dots on the plot.
For example, if the function was [tex]x^2[/tex], I would want to start with a table, and plug in my x-values to given formula as I go through each x.
x = -5 then y = [tex](-5)^{2} =25[/tex]
x = -4 then y = [tex](-4)^{2} =16[/tex]
x = -3 then y = [tex](-3)^{2} =9[/tex]
x = -2 then y = [tex](-2)^{2} =4[/tex]
x = -1 then y = [tex](-1)^{2} =1[/tex]
x = 0 then y = 0
x = 1 then y = [tex](1)^{2} =1[/tex]
x = 2 then y = [tex](2)^{2} =4[/tex]
x = 3 then y = [tex](3)^{2} =9[/tex]
x = 4 then y = [tex](4)^{2} =16[/tex]
x = 5 then y = [tex](5)^{2} =25[/tex]
Of course you don't have to use for the x-values -5 to 5 if it's too long. At least 5 points should be made on the graph to improve accuracy. Other things like max, min, y-intercept, slope, x-intercept, etc all matter when graphing but if you were to just look at how to draw the basic functions first and how they look like, everything else will come naturally.
Now that the table of x and y values are made, you just need to plot this on a graph and connect the dots. The rest of the graph can be sketched to be an estimate of what the other values of x would possibly be equal to. As long as the bottom points of [tex]x^2[/tex] is complete and accurate the rest is drawn as close as you can make it, if that makes sense...
Hope that helps!
