The Gradient Function Investigation

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The Gradient Function

Introduction

Curves on a graph can be of varying steepnesses. This steepness also varies from point to point on many graphs. The steepness of a curve at a point is called its gradient. There are several methods for calculating the gradient at a certain point on a curve including the 'Tangent Method' and the 'Small Increments Method'.

The Tangent Method

Calculating the gradient of a straight line is simple. The formula is:

Gradient = Change In Y

Change In X

This formula is demonstrated on Graph A. A curve proves more of a problem as the gradient is constantly changing. To calculate the gradient at a certain point, we must somehow be able to create a straight line from which to calculate this gradient. This can be achieved by drawing a tangent to the curve at the point in question. A tangent is a straight line which touches the curve at one point and one point only. Calculating the gradient of this tangent will give the gradient of the curve at this point. This is illustrated on Graphs B1, C1 and D1.

This method is not very accurate though and large discrepancies in gradient can occur. This is because this method involves manually drawing a tangent which will often be drawn incorrectly. Errors in the measurement of changes in Y and X can also occur, even on a very large scale graph.

The Small Increments Method

This method is more accurate than the 'Tangent Method' as it does not involve manual drawing and measurement of changes. This method works on the basis of drawing a chord (a line joining two points on the curve) from the point where the gradient will be taken to another point close by. As the function of the graph (i.e. y = x²) can be used to calculate the precise X and Y co-ordinates of the second point on the curve, the exact changes in X and Y can be calculated. This is achieved by subtracting the X and Y values of the point where the gradient is to be calculated from the X and Y values of the second point on the curve. This means the calculation of the gradient of the chord will be completely accurate.

Gradient of Chord = Change In Y = (Y + m) - Y

Change In X = (X + n) - X

(where m is the increase in Y and n is the increase in X)

A second chord, nearer to the point where the gradient is to be calculated can then be drawn and its gradient calculated. After a series of these chords has been drawn, the latter of which will be only a very small X value increase away from the point where the gradient is to be taken, the gradients can be examined. It should become obvious that they are getting nearer and nearer to a certain number and this number will be a reasonably accurate estimate of the gradient at the point in question. This method is illustrated in Graphs B2, C2 and D2. This method is only as accurate as the number of chords measured and the gradients calculated will never be exactly the same as the actual gradient required.

What is a Gradient Function?

A gradient function is a rule, specific to a certain graph (e.g. y = x²) 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. The purpose of my investigation will be to find a formula which gives the gradient function for any curve of the form:

y= Ax + Bx +C

The 'Small Increments of Size "h" Method'

This method is based on the 'Small Increments Method' but instead of increasing the X value 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. Once this gradient has been calculated, it will give the hypothetical gradient at the point (x + h)². This gradient (i.e. 7 + h) can be converted to the gradient at the actual point in question by reducing the 'h' value to 0 so the remaining equation is the gradient at this point. This method is illustrated on Graphs E and F.
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Using the 'Small Increments of Size "h" Method' to Calculate a Gradient Function

To use the method above, it is necessary to replace the specific X value with a hypothetical one also. For the graph y = x² it is calculated as follows:

Gradient = (x + h)² - x²

(x + h) - x

= (x² + 2xh + h²) - x² (expand brackets)

x + h - x

= 2xh + h² (cancel x²)

h

= 2x + h (cancel h)

As the h value then tends ...

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