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Coursework Notes - Algebra

 

Graphs - Functions

 

y= f(x) + k

y= f(x + k)

 

 

 

The function      y= f(x) + k

 

functions #1transparentfunctions#2

 

fx plus ktransparentf plus k #2

 

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 transparentfunctions #3

 

 

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The function      y= f(x + k)

 

functions #1transparentfunctions#7

 

fx plus ktransparentf-x plus 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:

 

functions #5

 

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 transparentfunctions#6

 

 

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The function      y = kf(x)

 

functions #1transparentfunction#8

 

kfx -1transparentk fx#2

 

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

 

 

Example - let k=5

 

functions#10

 

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.

 

 

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The function      y= f(kx)

 

functions #1transparentfunctions#9

 

kfx -1transparentf of 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

 

functions#12

 

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

 

functions#13

 

 

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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.)

 

functions#14

 

sin#1

sin#2

 

 

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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.)

 

cos#1

 

cos#2

 

 

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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)

 

function#14

 

sin#1

 

functions#15

 

sin#3

 

 

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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).

 

functions#15

 

cos#1

 

functions#17

 

cos#3

 

 

 

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