The Gradient Function

The Gradient Function Introduction The gradient of any line is the steepness at which it slopes; on straight lines it can be worked out by drawing a right angled triangle using the line itself as the hypotenuse to find out the ?y, and ?x. The gradient of a line can then be worked out by dividing ?y by ?x. The following graphic shows an example: However, with a curved graph, the gradient is different at different points. To work out the gradient at a point of a curved graph, a tangent would have to be drawn, and the gradient of it measured. The longer the tangent is, the more accurate the result if done by eye. The following graphic is an example: Because this method is inherently inaccurate, to improve the accuracy we could either use a computer program to draw an accurate tangent, or use the small increment method. The small increment method is where the gradient of a chord, from the point, to another point of the line a short distance away, is worked out to find the gradient between the two points. So, for example, if it was a curve of y=x2, the gradient at x=3 would be measured using a chord from [3, 9] to [3.0001, 9.00060001] and so the gradient would be 0.00060001 divided by 0.0001 which is 6.0001 or 6 as an integer. The Small Increment Method I given a curve, y=x2, and a point, x=2, I can calculate an approximate gradient by using a chord with a second point a

  • Word count: 2700
  • Level: GCSE
  • Subject: Maths
Access this essay

The Gradient Function

The Gradient Function I am trying to find a formula that will work out the gradient of any line (the gradient function) I am going to start with the most simple cases, e.g. y=x, y=x², y=x³ etc. They are probably going to be the easiest equations to solve as they are likely to be less complex, and hopefully the formulas to the more complex equations will be easier to discover by looking at these first formulas. I am going to look at the line y=x² first. y=x² x 2 3 4 y 4 9 6 Please refer to graph on separate piece of paper One of the most obvious things I notice is that as the co-ordinates increase so does the gradient. Not only can you see that from the results below, but also on the graph you can see that the line gets steeper and steeper. This makes sense, as the higher the number x is the larger the difference between x and x². Another thing that I have noticed is that the larger the co-ordinates the smaller the increase in gradient. Point Gradient (tangent) Gradient (Increment Method) (1,1) 2 2.01 (2,4) 3.3 3.01 (3,9) 6.36 (2dp) 6.01 (3.5,12.3) 6.4 7.001 As the table above shows there are two methods that I am using for calculating the gradient of line. The first being drawing a tangent at the point, working out the distances on the tangent using the scale on the graph and then using this formula: dy/dx Increment Method However

  • Word count: 1327
  • Level: GCSE
  • Subject: Maths
Access this essay

The Gradient Function

The Gradient Function 'x=a' is a vertical line which intercepts the x-axis at point 'a'. 'y=a' is a horizontal line which will intercept the y-axis at point 'a'. 'y=ax' and 'y=-ax' are the equations for a sloping line which intercepts at the origin. The value of 'a' is the gradient of the line, so therefore the larger the value of 'a' the steeper the gradient of the line. I am trying to find the gradient function this is a formula that will work out the gradient of any line. The equation to find the gradient of a straight line is: Gradient=y1 - y2 X1 - x2 The gradient of a straight line is a constant, the gradient of a curve is not. To find the gradient of a curve we need to choose certain points along the curve, e.g. at x=1, x=2...etc. We then draw a tangent from these points and work out the gradient of the tangent. I am going to start by looking at the equation 'y=axn' . When: a=1 n=1 then the equation is Y=x a=2 n=1 then the equation is Y=2x In the equation Y=ax¹ the gradient is 'a' Y=X2 I am going to look at the line y=x2 first: x -3 -2 -1 0 2 3 x2 9 4 0 4 9 y 9 4 0 4 9 (See graph, fig.1) From the graph we can see that as the values of the coordinates increases, so does the gradient. We can see this from the results below and the graph. Point Gradient (tangent) Gradient (small increment method) (1,1) 2 2.0001 (-1,1)

  • Word count: 471
  • Level: GCSE
  • Subject: Maths
Access this essay

The Gradient Function Maths Investigation

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: . 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) and (3.1,9.61). You connect the two points together using a straight line and because the graph is on a much larger scale the line should follow almost the same path as the curve. You then use the formula

  • Word count: 1374
  • Level: GCSE
  • Subject: Maths
Access this essay