How to draw graph f(x) from f'(x) (1 Viewer)

[girlworld_club]

these questions always pop up simultaneously in past hsc papers, however my current maths in focus books does not explain anything about how to do this. and i mean nothing. I am so stuck on these questions.

can anyone offer an explanation.

examples of questions include ones were they provide f'(x) graph and ask you to sketch what f(x) would look like. Please. i am sooooooo confused.

 

[Focus is Key]

Hopefully someone can explain this better than me.

What I always do is consider what the highest power of x will be, which will in turn tell how many bumps your graph will have. E.g. a y=x^2 graph will have one bump etc for will help with the direction of the graph. I usually place any stationary points on the x-axis and use them as a guide for the sketch. By the way, if the graph begins (i.e. the left side of graph) starts above the x axis, the differential will start below the x-axis.

 

[michaeljennings]

What I used to do was this

First find where the f'(x) graph cuts the x axis because this is where the turning points for f(x) occur

Then just look at the f'(x) and identify that wherever y>0 then f(x) has a positive gradient and should slope upwards, and wherever y<0 then f(x) has a negative gradient and should slope downwards

From there it should be simple to figure out how to draw it if you know where the turning points are

 

[Sy123]

These questions become quite simple if you understand what differentiation truly means.

If we differentiate a function, we get a function which tells us the gradient of the function that we differentiated.

So if we want to find the gradient of the function f(x) at x=2
We have to compute f'(2) which will give us the gradient/slope of the original function.

This is why f'(x) = 0 will give us stationary points, because if slope = 0 then its horizontal.

So using only the logic of differentiation we must go backwards from f'(x) to f(x)

So we must analyse the given f'(x) and look at the y-values of f'(x) since they will tell us the gradient of f(x).

If the y-values are negative, i.e. if f'(x) < 0 then the slope of f(x) will be downwards (decreasing)
If the y-values are positive, i.e. if f'(x) > 0 then the slope of f(x) will be upwards (increasing)
If y-values are zero, i.e. f'(x) = 0 then the slope of f(x) will be zero (stationary)

Remember, these are just examples to situations that we analyse f'(x) with, there could exist, stationary points, asymptotes, open holes, sharp points (within f'(x)) etc etc.
Just remember that f'(x) is the gradient function of f(x) and that should be all that you need.