Investigating Slopes

Alessandro Marazzani

25/11/13

Maths Assessment

Introduction

The aim of our assessment is to find general laws and correlations with the investigation of slopes. In fact, we had been given a sheet, where we had many points to analyse and check and, at the end, come up with a conclusion. We are allowed to use our graphic calculator, mainly because we need to show our results.

I will structure this essay by firstly doing an introduction, and then I will start by answering each point we have been given to analyse and come up with a statement at the end of each point. At the end, I will write a final conclusion that sums up all the results I found in this big investigation of slopes.

Investigation 1

The first investigation of this assessment tells me to find a formula for the gradient when there is the function f(x)= X2. It also tells me to write the answer in the form of f1(x). So, to do this, I will plot the function in the graph, check the gradient, and see if there is a formula for it. This investigation is quick to complete, so I recon, I don’t need to do anything to my results, but only find a connection between them.

The conclusion I can draw out from this investigation is that at the end, I found a correlation with the gradient of my slope. For every value I give to X, every time the gradient is double the value. In fact, I can say that when f(x)= X2, this means that f1(x)= 2X

Investigation 2

This investigation is much more longer than the first investigation. It is also more complex. It tells me to consider two functions, which are f(x)= xn and f(x)=aXn. then I will still need to find an answer for f1(x) for the function f(x)=aXn and then discuss the scope of my analysis. To make my life easier I will divide this investigation in parts. For every part, I will analyse a variety of cases, and come up with a conclusion at the end of each case and part. Consequently, after watching at all my results, I will give a final formula to this investigation.

Part 1

For the first part of the second investigation, I decided to first look at the function f(x)= xn. So, to work with this function, I will give to the input “n” the values of 3,4 and 5. I decided to not give to “n” the value of 2 because I have already analysed the answer of it in the first investigation. So, I will start by giving to “n” the value of 3.

1st Case

I will try, firstly, with “3” for the number that will equal to n. So, f(x)= X3. I have decided to start with number 3 because, as I wrote before, my first investigation already told me the answer if, as my first case, I would have made n=2.

At the conclusion of my first case for my second investigation, I found again one link. As my first investigation, I spotted the correlation after I analysed the gradients. In fact, I arrived at the conclusion of stating that if f(x)= X3, consequently f1(x) will be f1(x)= 3X2

2nd Case

I still continue by increasing the number of one, so that I will not miss any important result. I will try to find a correlation when n=4. Thus f(x)= X4

Again, with these point, I can say by looking at the gradient, that f1(x)= 4X3. Now, I’m starting to have a brief supposition of what my general discussion and results for investigation number 2 will be. However, to be sure, I will try to find a correlation by looking at one more case.

3rd Case

For my last case, I will let n=5. In fact, the formula on which I will investigate the gradient will be f(x)= X5 .

Finally, I have arrived at the conclusion for the first part of Investigation 2. However, before, arriving to this conclusion, I had to analyse my results of the 3rd case. In fact, I can state that when, in this case, f(x)= X5 it means that f1(x)= 5X4