• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17
18. 18
18
19. 19
19
20. 20
20
21. 21
21
22. 22
22
23. 23
23
24. 24
24
25. 25
25
26. 26
26
27. 27
27
28. 28
28
29. 29
29
• Level: GCSE
• Subject: Maths
• Word count: 2856

# The Gradient Function Maths Investigation

Extracts from this document...

Introduction

Introduction

A lot of graphs produce lines that are curves. Some of the curves are steep, and some are not.

In this investigation I will be looking to work out a formula, which will work out the gradient of any curve.

There are three methods for working out the gradient of a curve, all of them using a tangent:

• The Tangent Method
• The Increment method
• General Proof

This is the graph of y=x². I will find out the gradient of this curve, by using the three methods I mentioned above. I will use the point x=2 for this graph.

Tangent Method: The tangent method is already shown on the graph. I have drawn a line which touches the edge of the curve, when x=2. Once I have drawn the line I then turn it into a triangle, and look at the change in y, and the change in x. Finally I use the formula:

GRADIENT= VERTICAL (CHANGE IN Y AXIS)

HORIZONTAL (CHANGE IN X AXIS)

Therefore the gradient of the curve equals:

The only problem with the tangent method, is that it is only an estimation.

Increment Method: The increment method is where you plot two points (P and Q) on the curve, and the draw a line to join them. This

Middle

5

This is the graph of y=x5. I will continue to investigate the gradients of the graphs y=xn.

Increment Method: x=3

P(3,35) and Q(4,45)

=45-35

4-3

=781

P(3,35) and Q(3.75,3.755)

=3.755-35

3.75-3

=664.8

P(3,35) and Q(3.5,3.55)

=3.55-35

3.5-3

=564.4

P(3,35) and Q(3.25,3.255)

=3.255-35

3.25-3

=478.4

P(3,35) and Q(3.01,3.015)

=3.015-35

3.01-3

=407.7

P(3,35) and Q(3.001,3.0015)

=3.0015-35

3.001-3

=405.3

P(3,35) and Q(3.0001,3.00015)

=3.00015-35

3.0001-3

=405.03

From these results we can conclude that the gradient on an y=x, when x=3 is 405.

Instead of using general proof to work out the equation for this graph, I will use another method, that I have taught myself. It is called Binomial Expansion.

A binomial expansion is the result of a binomial expression, like (x+h) being raised to a power.

The simplest binomial expansion is (x+1), but I will show you the expansion of (x+h)5. So the expansion is:

1   x5 + 5  x  h + 10  x  h + 10  x  h + 5  x  h + 1  h

x5   +  5x4h    +  10x3h2  +  10x2h2 +   5xh4  +   h5

We can now, add the remainder of the formula and simplify this down, like in general proof:

x5+5x4h+10x3h2+10x2h2+5xh4+h5-x5

h

=h(5x4+10x3h+10x2h+5xh3+h4)

h

=(5x4+10x3h+10x2h+5xh3+h4)

=5x4

Therefore when:

Conclusion

n-1, into two.

For example, we shall take the graph, 2x²+3x³:

As we can see, two terms involved, therefore if we were to use our formula naxn-1, it would not work, as there is more than one term. Therefore we have to add something to the formula, to include the second term. So when there are formulas that have more than one term involved, the formula is:

naxn-1+laxn-1

Here the change is, the inclusion of laxn-1. This of course is the second term. The formula would, change depending on the amount of terms involved.

So therefore form this formula I can work out the gradient of the graph y=2x²+3x³.

Let’s first take the graph 2x². I, previously got the formula 4x. The second graph 3x³, we haven’t looked at before, but with our formula, we can see that it is equal 9x².

This means that to find the gradient of the curve y=2x²+3x³, we use the formula 4x+9x².

Therefore the gradient of the curve when:

x=1 is 4+9=13

x=2 is 8+36=44

x=3 is 12+81=93

x=4 is 16+144=160

If I were to work out a similar graph to, y=2x²+3x³, using differentiation, I would use exactly the same method. I would take each term in turn, and work out their respective formula, thus enabling me to put this into the general formula.

This student written piece of work is one of many that can be found in our GCSE Gradient Function section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Gradient Function essays

1. ## The Gradient Function Coursework

5 star(s)

But let us go back to the chord method. This method will never be accurate, as we could never calculate the gradient of the chord which joins two points, which have the distance zero. The formula m=?y / ?x would not work for this case (? is a Greek letter and just means 'difference', so the notation literally means

As you can see by looking at the graph, another set of co-ordinates have been plotted to discover the change in x and y on the tangent at x=-1. As you can see from the graph, the change in x is 0.5, and the change in y is once again 2.

1. ## The Gradient Function Investigation

y = x�) which can be used to precisely calculate the gradient at any point on the X - axis. This rule will be applied to the X value in question to give an exact gradient value for this point.

2. ## Maths Coursework - The Open Box Problem

For example for the 40:20 cut out the equation would be: y = x(40-2x)(20-2x), where y is the volume. I have put y = because you cannot form a table by saying v = is Autograph. Again like the previous task I have made the graph bold so that it

I got to the conclusion that my slope is 2x by seeing the relationship between my x values and my gradients. When my fixed value was 5, the gradient was 10 and when I divided them I got 2. Also when my x value was 2, my gradient was 4 and when I divided them I got 2 again.

32 x y change in y change in x gradient 2 17 15 1 15 2.1 18.23 13.77 0.9 15.3 2.2 19.52 12.48 0.8 15.6 2.3 20.87 11.13 0.7 15.9 2.4 22.28 9.72 0.6 16.2 2.5 23.75 8.25 0.5 16.5 2.6 25.28 6.72 0.4 16.8 2.7 26.87 5.13 0.3 17.1