• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

The Gradient Function Maths Investigation

Extracts from this document...


The Gradient Function I am going to be investigating the function of the gradient. A function is a variable that depends on the value of other independent variables and the gradient is the steepness of a line or curve. I am going to try to work out a formula that will calculate the gradient of any given line or curve; this will be the gradient function. I already know of some methods that can be used to calculate the gradient, these are: 1. The formula: increase in y increase in x This formula represents the vertical value on the graph divided by the horizontal value. It can also be written as: dy dx This notation demonstrates the rate of change at y with respect to x. Which means that as x changes so does y. When using this formula to work out the gradient at of a curve a tangent must first be drawn. 2. 'Omnigraph' is a computer program which will create a graph for a given a formula. It will then draw tangents on the graph and work out the gradient. 3. The small increment method can be used as a more accurate way of calculating the gradient of a graph. To use it you must zoom in on a section of the graph, for example the coordinates (3,9) ...read more.


going to put the formulas I have discovered so far into a table to make them easier to interpret: Equation of graph Gradient Function y=x� 2x1 y=x� 3x� y=x4 4x� n nx(n-1) From the formulas I have found using my graphs I can see the general rule and from them I have worked out this formula: Gradient= nx(n-1) This formula is correct for all the graphs I have done so far but I am now going on other graphs. Firstly I am going to see if it works for negative numbers on the graph y=x -1 (see graph 5). These are the results I have found using this graph. I have also tested them to see if the formula Gradient= nx(n-1) works: x Gradient=nx(n-1) Actual Gradient 1 = -1x1(-1-1) = -1 -1 2 = -1x2(-1-1) = -0.25 -0.25 3 = -1x3(-1-1) = -0.11 -0.11 4 = -1x4(-1-1) = -0.0625 -0.063 This graph is evidence of my formula for graphs with equations with negative integers but not with all numbers. I am now going to try it with a graph with a fraction integer, y=x1/2 (see graph 6). These are the results I have found using this graph. I have also tested them to see if the formula Gradient= nx(n-1) ...read more.


After again consulting the A level textbook I have found that as H is worth such a tiny amount that it does not matter. This is because the gradient of Q is on the line of PQ and is therefore equal to the gradient of P. This means H is worth virtually nothing. Therefore: Gradient of tangent at P =2x dy =2x dx These workings proves that my formula, Gradient= nx (n-1) is correct for the graph y= x�. I am now going to try to prove my formula for the graph y=x� using this method. I have drawn a sketch graph (graph 8) that is very similar to the graph previously used. The symbols all represent the same values as before. (y + K) = (x + H)� (y + K) = x� + 3Hx� + 3H�x + H� K = 3Hx� + 3H�x + H� Gradient of PQ = K H = 3hx� + 3H�x + H� H = 3x� + 3Hx + H� The Hs can now be ignored leaving: dy =3x� dx As I predicted this method shows that the gradient of the graph y=x� is 3x�. I have discovered that the gradient of all the graphs I have used can be found using the formula nx (n-1). This formula works for positive, negative and fractional integers. I have also gone some way to proving my formula algebraically by using differentiation. ?? ?? ?? ?? Sarah Shea 11A Maths Investigation ...read more.

The above preview is unformatted text

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

See related essaysSee related essays

Related GCSE Gradient Function essays

  1. Peer reviewed

    The Gradient Function Coursework

    5 star(s)

    'the difference in y over the difference in x equals the gradient). The result would be 0/0 which is undefined and would not give a sensible answer. We call this the limit, and we cannot calculate this. What we can do though is to make the difference between the two

  2. 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.

  1. Investigate gradients of functions by considering tangents and also by considering chords of the ...

    zero. We can rewrite the coordinates of (x, y) as (x, f(x)) and the coordinates of (x + dx, y + dy) as (x + dx, f(x + dx)), since y is a function of x (y = f(x)). 9 So the gradient of the curve is: lim (y + dy - y)

  2. The Gradient Function Investigation

    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

  1. Gradient function

    0.1 113.521 3.2 104.8576 23.8576 0.2 119.288 3.3 118.5921 37.5921 0.3 125.307 3.4 133.6336 52.6336 0.4 131.584 3.5 150.0625 69.0625 0.5 138.125 3.6 167.9616 86.9616 0.6 144.936 3.7 187.4161 106.4161 0.7 152.023 3.8 208.5136 127.5136 0.8 159.392 3.9 231.3441 150.3441 0.9 167.049 3.99 253.4496 172.449584 0.99 174.1915 3.999 255.7441 174.744096

  2. Gradient Function

    is 5, 125 x y change in y change in x gradient 4 64 61 1 61 4.1 68.921 56.079 0.9 62.31 4.2 74.088 50.912 0.8 63.64 4.3 79.507 45.493 0.7 64.99 4.4 85.184 39.816 0.6 66.36 4.5 91.125 33.875 0.5 67.75 4.6 97.336 27.664 0.4 69.16 4.7 103.823 21.177

  1. The Gradient Function

    value of x:- Tangent at x 1 1.5 2 3 -2 Gradient 4 6 8 12 -8 Now we can predict the gradient function to be 4x as the gradient is always 4 times of x. For the curve y=3x�, the gradient of the tangent at x is six times

  2. The Gradient Function

    the line is: Y2-Y1 X2-X1 = 4-4 = 0 2-1 Graph: Y= -0.25x0 The table below shows the points, which I have used to plot the graph: X Y -1 -0.25 0 -0.25 1 -0.25 The gradient of the line is: Y2-Y1 X2-X1 = -0.25 - (-0.25)

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