1. ## Arc Lengths Question

Okay so I know how to draw my functions
but as soon as we get into a function such as: x^2 +2x , or something like that. I get confused. My professor explained that, using the first and second derivatives of functions such as these, they can help us plot. I'm putting this under arc lengths, because this is where im having this problem. So if someong can help me with drawing more complex similar to this example, then that would be great!

4. The arclength is very complicated and won't help you draw the graph of $\displaystyle y= x^2+ 2x$. y= x(x+ 2) so the graph has y-intercepts at (0,0) and (-2, 0). Of course, (0, 0) is also the x-intercept. y'= 2x+ 2= 2(x+1)= 2(x-(-1)). If x< -1, y'< 0 and the graph is decreasing. If x> -1, y'= 0 and the graph is decreasing. y"= 2> 0 so the graph is always concave upward. The minimum value occurs when x= -1 and is y=(-1)(-1+2)= -1. Since the function is quadratic, the graph is a parabola, opening upward with vertex at (-1, -1).