# Math Help - drawing the original graph from the gradient graph given.

1. ## drawing the original graph from the gradient graph given.

The following is a gradient graph of a walking trail. I am asked to draw it's original distance-height graph. I have attempted to draw this and i think i am correct. I just need clarification.

Will the original graph be height as the y axis and distance as the x axis?

the above is the my attempt of drawing it. It's rough but without looking at the graph, is it true that at the turning point of the gradient graph, the slope on the original graph should be the largest, so just after the distance is 1000 meters the graph will be steep, much like a point of inflection. Than when the gradient graph crosses the distance axis, the slope of the distance-height graph is 0, meaning that just before 2000 there is a turning point. Furthermore, once the gradient graph passes into the negative y region the slope of the original graph becomes negative and again the graph will reach a steep point just after 2000m but not as steep as the one just after 1000m. It then comes back up to intersect the distance axis, so there must be a turning point on the original graph before 3000m, indicating that there is a change in gradient and the gradient is now positive, it reaches a steep point once again just a tad after the turning point before 3000m, but this point is not as steep as the previous 2, finally the trail concludes with a turning point just after 3000m, trailing downhill and reaches the highest steepness of it's remaining meters just before 5000, then flattens out at 5000m.

Is this correct?

P.s, i know that the last part of my original graph drawing is imprecise but it's just a rough sketch.

2. ## Re: drawing the original graph from the gradient graph given.

Hey johnsy123.

The peaks and troughs for your final function should occur when the derivative is equal to 0. So you will have a maximum at the first point when the derivative = 0, a minimum at the next point the derivative = 0, a maximum after that and so on (all are local minimums and maximums).

You've done the above in the graph and really the rest of the details is getting fine-control of the shape but that's not really easy in all situations.

3. ## Re: drawing the original graph from the gradient graph given.

Hello, johnsy123!

I am making approximations for some of the vital points of the graph.
And I'll make comments based on those approximations.

The following is a gradient graph of a walking trail.
I am asked to draw it's original distance-height graph.

The gradient is zero at the start (level ground).
Then the trail gets steeper and steeper until $x = 1200.$
Then it gets less and less steep (still going uphill) until $x = 1800.$
At $x = 1800$, the trail is horizontal.
We have reached a maximum point.

The trail goes downhill, getting steeper and steeper until $x=2200$
Then the trail gets less and less steep (still going downhill) until $x = 2600.$
At $x = 2600$, the trail is horizontal.
We have reached a minimum point.'

The trail goes uphill, getting steeper and steeper until $x = 2900.$
Then it gets less and less steep (still going uphill) until $x = 3200.$
At $x = 3200$ the trail is horizontal.
We have reached a maximum point.

The trail goes downhil, getting steeper and steeper until $x = 4500.$
Then it gets less and less steep (still going downhill) until $x = 5000.$
At $x = 5000$, the trail is horizontal.
And we reach a minimum point at the end of the trail.