calculusgraphing-functionslimits
Sketch the graph of an example of a function f(x) that satisfies all of the following conditions:
Here is what I have so far:
Am I on the right track? I think the graph satisfies all of the conditions, but the lines cross at about (2,3)- is that acceptable? I know there are probably a large number of ways to draw this, is there a better way I should be aware of?
How does this look? Does the graph now satisfy the conditions?
Your sketch doesn't even look like the graph of a smooth function - remember that it should just be a deformed plane with nowhere vertical tangents. Here's an animation I whipped up that might help your intuition for this example:
In general it's helpful to remember how changes in the topology of the level curves correspond to local features of the function: a loop appearing/disappearing corresponds to a local extremum, while a transition across a self-intersection like you see in this example at $c=0$ corresponds to a saddle point.
zeros of $f$ are values $c$ such that $f(c) = 0$.
I believe there is a problem with your graph at $x = -3$.
Your graph is not defined on $(2,\infty)$, so I don't believe it is continuous on $(0,\infty)$
This is what I am getting:
Those should be negative numbers on the left. Hehe.
Best Answer
You also need to mark the asymptotic points at $(0,-2)$ and $(0,1)$: