# Gradient Graphs Investigation

Extracts from this document...

Introduction

Derek Pollard 5M Page

I have been set a task to find the gradient of different graphs ie. The gradient of y = x² or y= x³.

I will start with the graph of y = x². (This is shown below)

I have found the gradients for all of the numbers shown, I did this by drawing a triangle as close to the graph line as I could. I would then find the height and the width of the triangle. Then I would divide the height by the width.

Here is an example:-

Height of triangle 3 = 12

Width of triangle 3 = 2

So.

12 = 6

2

The gradient of point (3,9) is therefore 6.

Here is a table of all of my results:-

Diff 1 2 2 2 2 2

This means that my formula should be:-

Gradient = 2x

I will now show the gradients of chords starting at point (2,4) and finishing at various other points along the plotted line ie. 2 to 5 and 2 to 4 etc.

Middle

2x is the gradient function of the curve y = x².

This can also be written as grad x² = 2x.

Here are some examples for when y = x²:-

When x = 5 y = 5² = 25

The gradient function is 2x.

The gradient = 10 when x = 5

The point is (5,25) and the gradient is 10.

I have checked this result and it is correct.

will now investigate the gradient of y = x³. A graph to show this is below:-

I have found out the gradient for each point on the graph. I did this in the way that I explained on page1. A table shows this below:-

Diff 1 3 9 15 21 27

Diff 2 6 6 6 6

This means that my formula for the gradient is:-

3x²

Here is a table for the chords:-

Diff 1 5.75 5.25 4.75 4.25 3.75

Diff 1 0.69 0.67 0.65 0.63

As you can see there is a pattern emerging from the chords of these numbers.

Below is another copy of the graph y = x³.

Conclusion

Multiply by the index and reduce the index by 1

This means that:-

Grad x = 5x

Grad x = 6x

## So my general formula will be:-

Grad x = nx

I know from previous class investigations into straight line graphs ie. y = 6 and y = -3 that these graphs have a gradient of zero. So Grad 4 = 0 and Grad –3 = 0.

So if we differentiate a constant we get 0.

So far this agrees with my general formula as x° =1. We can write this as Grad 4 = Grad 4x ° = 0 × 4x = 0.

Again I know that graph y = mx + c has a gradient of m and graph y = x has a gradient of 1 and graph y = 6x has a gradient of 6.

So Grad x = 1

Again this agrees with my formula as:-

Grad x = 1 × x° = 1

and,

Grad 6x = 6 × grad x = 6 × 1 = 6.

This shows the general fact that if a function has a factor which is constant, that constant remains the same as a factor of the gradient function.

This concludes my coursework.

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

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