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Gradient Function - To discover different curves and their relations, and the formula for the gradient.

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Introduction

Gradient Function Coursework

Aim

To discover different curves and their relations, and the formula for the gradient.

What is a gradient?

My understanding of a gradient is that it tells us how steep a slope is. The Oxford dictionary says ‘1 a sloping part of a road or railway. 2 the degree of a slope, expressed as change of height divided by distance travelled. 3Physics a change in the magnitude of a property (e.g. temperature) observed in passing from one point or moment to another.’.

How will I find the gradient?

I will find the gradient by using two methods; the tangent method and the increment method.

The tangent method involves drawing a right angled triangle with the hypotenuse touching the point you want to find the gradient of, and stretching to the two other corners.

See the graph ‘Tangent Method’ .

On this graph we see four letters, D being that of the point and A, B and C being of the triangle. Now to find the gradient with this method we do BC÷AC.

...read more.

Middle

X Value

Gradient (g)

Tangent Method

Increment Method (to 2 d.p.)

1

3.03

2

12.06

3

27.09

4

48.12

As you can see my predicted formula did not work. Also, as the lines are getting more and more steeper the tangent method is becoming increasingly inaccurate.

My predicted formula g=3x produces a result which I can use to figure out a formula; every time (excluding the first answer) I multiply x by 3 I get a number, and if I square that number I get the gradient. This also proves that the increment method does work.

So therefore the discovered formula is g=3x2

Therefore, I predict that the formula for y=x4 will be g=4x2

Y=x4

As the tangent method is becoming increasingly inaccurate I am just going to use, the already proven to work, increment method.

Please turn over

X Value

Gradient (g)

Increment Method (to 2 d.p.)

Formula Answer

1

4.04

4

2

32.24

16

3

108.54

36

4

256.961

64

As you can see from above, my formula does not work. But there is a pattern somewhere so I

...read more.

Conclusion

2.

Y=x-1

X Value

Gradient (g)

Increment Method

Formula Answer

1

1

1

2

0.25

0.25

3

0.11

0.111…

4

0.06

0.0625

The method has been successful.

When trying y=x-1/2 I encountered various errors with the calculations so I have decided to leave that one.

More Complex Equations

So far, I have only covered numbers where there is an x to a power. Now to show that the formula works with more complex equations that have more parts than just the xx.

y=2x2

This is more simple than it actually seems. The method for working out this one involves splitting up the equation so that I work out the y=x2 first in the way I have explained with the formula, and then just multiply the output of the formula by the number next to x , which in this case is 2.

y=2x2+3x

This is also the same as above, you split the equation up. I would split up the y=2x2 and work that out using the method above. Then I would take the +3x and work it out as y=3x. After the gradient has been attained for those two parts, I would then add it together and then we would have our gradient.

...read more.

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

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