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**Coursework Notes - Algebra**

**Graphs - Functions**

y= f(x) + k |
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__The function__ **y= f(x) + k**

When x = 0, y = k . So the curve is moved(translated) by 'k' in the y-direction.

In vector terms the translation of the curve is

__The function__ **y= f(x + k)**

This is best understood with an example.

Let k be equal to some number, say 3. Adding 3 into the original equation, we have:

So the curve moves -3 to the left, to where y=0. That is -k to the left.

In vector terms the translation of the curve is

__The function__ **y = kf(x)**

In our example, y increases by a factor of 'k' for every value of x.

__Example__ - let k=5

So for each value of x, the value of y is 5 times its previous value. The curve is stretched in the y-direction by a factor of 5. That is by a factor of k.

__The function__ **y= f(kx) **

In the above, when x=1, y=1. However, in the second function when x=1, y is a higher value.

Look at the example below for x=1 and other values of x.

Remember, in this function the constant 'k' multiplies the x-value **inside** the function.

__Example__ #1 - let k=4

You will notice that the y-value jumps by a factor of 16 for each increasing x-value. The y-value increases by a factor of 4 squared.

With more complicated functions the value of y for a given value of x, increases once more, narrowing the curve in the x-direction(or stretching in the y-direction).

__Example__ #2 - a more complicated function with k=4

__The function__ **y= sin(x+k)**

Here the graph is translated by the value of k, to the **left **

So when k=90 deg. The curve moves horizontally 90 deg. (looking at the red dot, from 270 deg. to 180 deg.)

__The function__ **y= cos(x+k)**

This is exactly the same as for the sine function.

The graph is translated by the value of k, to the **left **

So when k=90 deg. The curve moves horizontally 90 deg. (looking at the red dot, from 180 deg. to 90 deg.)

__The function__ **y= sin(kx)**

Here the graph is squeezed horizontally(concertinered) by a factor of k.

In our example below, k = 2. So one whole wavelength of 360 deg. is reduced to 180 deg.

Conversely you may think of any value of x being halved(red spot reading changes from 270 deg. to 135 deg)

__The function__ **y= cos(kx)**

As with the previous function, the graph is squeezed horizontally(concertinered) by a factor of k.

In our example below, k = 2. So one whole wavelength of 360 deg. is reduced to 180 deg.

Conversely you may think of any value of x being halved(red spot reading changes from 180 deg. to 90 deg).

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