## Equation of a Line: y = mx + c

The equation of all straightlines can be written in the form

**y=mx+c**

where m is the gradient, c is the intercept on the y-axis and you are plotting xagainst y. m and c ae constants meaning their value is fixed.

The y intercept, c, is how far up, or down if its negative, he line crosses the veritcal yaxis.

The gradient, m, is how step the line is. A gradient equal to 1 means that the graph is at 45 degress to the axis, a gradient greater than 1 is steeper and a gradient less than 1 is shallower.

To find where the graph cuts the x-axis simply let y=0 and find the value of x since at the x-axis the grap has zero height so y=0.

The easiest way to sketch a graph of the form y=mx + c is to find the x and y intercepts and then draw a straight line through these points.

### Find the gradient of a line

The gradient, m, ofÂ a straight line can be found using the equation

m=change in y/change in x = (y_{1}-y_{2})/(x_{1}-x_{2})

Where x1 and y1 are the co-ordinates of one point on the line and x2 and y2 are the coordinate of another point on the line.

If you have any comments, questions or improvements please them as a comment below.

**By David Woodford**

## Sine and Cos Graphs Differentiating sin and cos

This is the basics of the sine cos and tan graphs and how sine and cos relate to give you tan. It also shows how to differentiate sin and cos.

The output or range of both sine and cos is from -1 to 1 when given any angle. They can be shown on a graph where y = sin(x) and y = cos(x). In these graphs all the angles go along the x axis and you can see a wave type shape is formed

**Sine Graph**

**Cosine Graph**

As you can see both the sin and cos graphs move periodically between -1 and 1 as the angles change, this pattern continues indefinitely because once you pass 360 degrees or 2 pi radians you will return back to the beginning. If you try to perform sin^{-1} of a value out side the range -1 to 1 you will get an error.

**Differentiate Sin and Cos**

also notice that the gradient of the sin graph is the value of the cos graph for the same angle and that the gradient of the cos graph is the -value of the sin graph for that angle. This means that we can differentiate the sin and cos graphs:

if f(x) = sin(x) then f ‘ (x)=cos(x)

and

if f(x) = cos(x) then f ‘ (x) = -sin(x)

however if we use ax instead of x we must differentiate it by bringing the a out, when its just x this doesn’t matter as the differential of x is 1.

ie)

let y = sin(f(x))

now let u = f(x)

du/dx = f ‘ (x)

also

y=sin(u) as u = f(x)

dy/du = cos(u)

from the chain rule

dy/dx = du/dx * dy/du

therefore

**if y = sin(f(x))
dy/dx = f ‘ (x)cos(f(x))**

**and similarly for cos
if y = cos(f(x))
dy/dx = -f ‘ (x)sin(f(x))**

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