# Thread: Graph of a function Question

1. ## Graph of a function Question

How can you sketch a graph of the function f that satisfies these conditions:

f(0) = 0

f"(x)>0 for x not equal to 0

lim f'(x) = ∞ , lim f'(x) = -∞
x →
0 - .............. x → 0+

lim f(x) = -, lim f(x)=
x
→-∞ ..... x→

2. ## Re: Graph of a function Question Originally Posted by Roundabout How can you sketch a graph of the function f that satisfies these conditions:

f(0) = 0 ...this means the graph passes through (0, 0)

f"(x)>0 for x not equal to 0 ...this means that the rate of change of the gradient is positive ie the gradient is increasing ie the graph is concave upward for x not equal to 0

lim f'(x) = ∞ , lim f'(x) = -∞
x →
0 - .............. x → 0+

... this means that as x approaches 0 from the left, the gradient gets steeper in a positive sense ie the graph almost becomes vertical. (the y axis (line x=0) will act like an asymptote.
... Similarly as x approaches 0 from the right, the gradients becomes more and more steeper in the negative sense.

lim f(x) = -, lim f(x)=
x
→-∞ ..... x→

... this means that as x moves further and further to the left (ie gets more and more negative, if you like) y moves further and further down. So the left hand side of the graph will be in the 3rd quadrant (think (neg big number, neg big number)). Opposite as x moves further and further to the right, y gets bigger and bigger, so the graph will end up in the 1st quadrant (think, (pos big number, pos big number)).
Try to draw the graph of y=f(x) now.

3. ## Re: Graph of a function Question

Thanks for the help. Wouldn't condition 2 (concave up) conflict with condition 4?

4. ## Re: Graph of a function Question

Not necessarily. As x approaches 0+, f(x) must approach infinity. So you have a JU shaped graph (if you can picture that).

5. ## Re: Graph of a function Question MMmm still not quite getting it. Maybe need to sleep on it some. 6. ## Re: Graph of a function Question

Starts at the bottom left corner(south west) in Quad 3, scoops up along the left hand side of the y-axis.
Dot at (0, 0).
"Parabola" shape on right hand side of y-axis, left side approaches y-axis like an asymptote, right hand side heads up north-east.
(not a very mathematical explanation). Draw it and you'll see it meets all the criteria.

7. ## Re: Graph of a function Question

So does this look right? Except I drew the left side of the parabola too close to the Y axis... 8. ## Re: Graph of a function Question

As x approaches 0+, f(x) must approach infinity.
This is not necessarily true. In fact, the following specific function is continuous everywhere. Also the graph that you drew can not be a parabola for x > 0, since we need the limit of the slope of f(x) as x approaches $0^+$ to be $-\infty$. 