• 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
• Level: GCSE
• Subject: Maths
• Word count: 1677

# In this project I will be using 2 methods to find the gradients of curves with the formula: y=axn

Extracts from this document...

Introduction

## Introduction

In this project I will be using 2 methods to find the gradients of curves with the formula:

y=axn

The two methods I will be using are the tangent method and the small increase method.

The tangent method involves drawing a line, by hand, which reflects the gradient.

The graph shown above is the curve y=x2. The line drawn on the graph is the gradient at the point (3,9). The Gradient is worked out by using this formula:

## QN

MN

In this example the gradient is:

9

1.5

= 6

This method is inaccurate because it is drawn by eye and no real mathematics was involved to find the gradient.

## The small increasemethod is a more precise method of finding the gradient of a curve. It uses 2 points on the same curve, which are very close to each other, and uses the straight line between the points as the tangent. As this increase becomes gradually smaller, the line reflects the gradient more and more.Graph → y=3x3

### Small increase method

Fixed point A (1,3)

B1 = (1.1,3.993)

B2 = (1.01,3.090903)

B3 = (1.01,3.009009003)

AB1 Gradient        =        3.993-3

1.1-1

=        0.993

0.1

=        9.93

Middle

B1= (3.1,3.13)

B2= (3.01,3.013)

B3= (3.001,3.0013)

AB1 Gradient        =        3.13-3

3.1-3

=        2.791

0.1

=        27.91

AB2 Gradient        =        3.013-3

3.01-3

=        0.2709

0.01

=        27.09

AB3 Gradient        =        3.0013-3

3.001-3

=        0.027009

0.001

=        27.009

The gradient is approaching 27

Tabulating results for the graph → y=x3

 X Y Gradient 1 1 3 2 8 12 3 27 27

As you can see, from this table and the results from the tangents of y=x3, that the tangent method is fairly inaccurate. The formula for this graph is:

## Graph → y=x2

### Tangent method

Point x= (1,1)

Point y= (2,4)

Point z= (3,9)

Triangle A        =        1

0.5

Triangle B        =        2

0.5

Triangle C        =        3

0.5

## Graph → y=x2

### Small increase method

Fixed point A= (1,1)

B1= (1.1,1.12)

B2= (1.01,1.012)

B3= (1.001,1.0012)

AB1 Gradient =        1.12-1

1.1-1

=        0.21

0.1

=        2.1

AB2 Gradient        =        1.012-1

1.01-1

=        0.0201

0.01

=        2.01

AB3 Gradient        =        1.0012-1

1.001-1

=        0.002001

0.001

=        2.001

The gradient is approaching 2.

Graph → y=x2

### Small increase method

Fixed point A = (2,4)

B1= (2.1,2.12)

B2= (2.01,2.012)

B3= (2.001,2.0012)

AB1 Gradient        =        2.12-4

2.1-2

=        0.41

0.1

=        4.1

AB2 Gradient        =        2.012-4

2.01-2

=        0.0401

0.01

=        4.01

AB3 Gradient        =        2.0012-4

0.001-2

=        0.004001

0.01

=        4.001

## The gradient is approaching 4.Graph → y=x2

Small increase method

Point A= (3,9)

B1= (3.1,3.12)

B2= (3.01,3.012)

B3=(3.001,3.0012)

AB1 Gradient        =        3.12-9

3.1-3

=        0.61

0.1

=        6.1

AB2 Gradient        =        3.12-9

3.01-3

=        0.0601

0.01

=        6.01

AB3 Gradient        =        3.0012-9

3.001-3

=        0.006001

0.001

=        6.001

The gradient is approaching 6.

Tabulating The Results For The Graph → y=x2
 x y Gradient 1 1 2 2 4 4 3 9 6

Conclusion

## Graph → y=2x3

### Small increase method

Fixed point A= (2,16)

B1= (2.01,16.241202)

B2= (2.001,16.024012)

B3= (2.001,16.00240012)

AB1 Gradient        =        16.241202-16

2.01-2

=        0.241202

0.01

=        24.1202

AB2 Gradient        =        16.024012-16

2.001-2

=        0.024012

0.001

=        24.012

AB3 Gradient        =        16.00240012-16

2.0001-2

=        0.00240012

0.0001

=        24.0012

The gradient is approaching 24.

## Graph → y=2x3

### Small increase method

Fixed point A= (3,54)

B1= (3.01,54.541802)

B2= (3.001,54.054018)

B3= (3.0001,54.00540018)

AB1 Gradient        =        54.541802-54

3.01-3

=        0.541802

0.01

=        54.1802

AB2 Gradient        =        54.054018-54

3.001-3

=        0.054018

0.001

=        54.018

AB3 Gradient        =        54.00540018-54

3.0001-3

=        0.00540018

0.0001

=        54.0018

The gradient is approaching 54.

## Tabulating The Results Fro The Graph → y=2x3

 x y Gradient 1 2 6 2 16 24 3 54 54

The formula for this graph is:

## y=axn – The Formula

Once this formula is found, you can work out any gradient function gradient. The formula I put forward is:

This is when:

N= the power of x (e.g. x3)

X= the x coordinate on the graph that relates to the gradient.

I shall test this formula with y=x2 and y=x3 curves.

Y=x2 2x(2-1) 1 x 2x1 2x y=2X

Y=x3 3x(3-1) 1 x 3x2 3x2y=3x2

When we look back at the result tables for y=x2 and y=x3 the formulas match. Therefore I conclude that the formula:

Is the correct formula and can be applied for all gradient functions.

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 the key idea is that the sum of the areas of the rectangular pieces will be a very close approximation of the actual area, and the more pieces we cut, the closer our approximation will be. At one point the approximation will be so close that the difference with the real volume would not matter anymore.

2. ## Curves and Gradients Investigation

by a specific number, it is instead increased by a hypothetical value, 'h'. The Y value is therefore increased to the value given when the function of the graph (i.e. y = x²) is applied to the X value: x + h.

1. ## Investigate the gradients of the graphs Y=AXN

To back up this evidence I am now going to work out the gradient function for two more examples were N= a negative integer. Y=X-3 X Predicted gradient with formula (3dp) X Co-ordinates Y Co-ordinates Increment formula Increment gradient 1 -3*1(-3-1)

2. ## I have been given the equation y = axn to investigate the gradient function ...

will now test for the line y = -2x+5 ,and according to the formula the gradient should be equal to -2 After drawing the graph for y=-2x+5 I have arrived at the following results: Y=-2x+5 Gradient function -2 I have arrived at the conclusion that the gradient function is =

1. ## I am going to investigate the gradients of different curves and try to work ...

Gradient = difference in y values = 16 difference in x values x-axis value 1 2 3 4 Gradient 4 8 12 16 I predict that the gradient for x = 5 will be 20. Test: (5,50) Gradient = 20 I used the formula 4x.

2. ## The Gradient Function Investigation

h = 2(x� + 3x�h + 3xh� + h�) - 2x� (expand and simplify) h = (2x� + 6x�h + 6xh� + 2h�) - 2x� (expand and simplify) h = 6x�h + 6xh� + 2h� (cancel 2x�) h = 6x� + 6xh + h� (cancel h) as h tends to 0 GF tends to 6x� 2.

1. ## The Gradient Function

As these are the same as when the tangent is at x=1, we can safely say that the gradient will be the same, but with one modification. As the tangent at x=-1 is travelling in a downward position, and the co-ordinates are located in the upper left quadrant of the graph, it will give us a negative gradient.

2. ## The Gradient Function

Graphs, which I am going to use: Y= -2x0 Y= 4x0 Y= -0.250 Y= 4x1 Y= 0.5x1 Y= -3x1 Y= 1/x Y= 3/x Y= -2/x Y= -0.5x2 Y= 2x2 Y= x2 Y= -0.5x3 Y= -2x3 Y= x3 Graph: Y= -2x0 The table below shows the points I have taken to plot the graph.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to
improve your own work