Constructing new functions by using elementary operations

Addition of Functions

http://www.youtube.com/watch?v=N89Cuq92e6M

Subtraction of Functions

http://www.youtube.com/watch?v=ZeW7X-ZTz2o

Multiplication of Functions

http://www.youtube.com/watch?v=ayFKhoAkuEk

Division Functions

http://www.youtube.com/watch?v=Wy5dOXWfaUs

Composition of Functions

http://www.youtube.com/watch?v=VavmbMh0HOI

How the graphs of functions change under different elementary operations of functions

How the graphs of functions change when subjected to the operations of 1) Addition/Subtraction of functions and 2) Multiplication/Division of functions

1) First, we look at what happens during addition and subtraction of functions.

Now to start off we can look at an easy example:

f(x) = 3x
g(x) = 4x
(f+g)(x) = 7x

We know from the coefficient of 7x which is 7 that this graph has become steeper; it basically got a steeper slope.

Let's try adding quadratic functions.

f(x) = $x^{2}$ and g(x) = 3 $x^{2}$ +5x

(f+g)(x) = 4 $x^{2}$ +5x

We know from the manipulations we can do to the graph of a function that having a higher number as the coefficient of a squared variable gives a tighter parabola so the graph then becomes skinnier. The 5x component also has an effect, but is less pronounced. The focus is on the leading coefficient (the coefficient of the highest degree component, in this case 4x^2).

From here we can then say that adding functions serves to 'squeeze' the function as it increases the leading coefficient (unless of course the functions are both constants, in which case the horizontal line simply changes its vertical position).

For subtraction, the same concept applies, only the coefficients are usually lowered (the word 'usually' is used here because subtracting a negative number is effectively adding it, thus increasing the coefficient). Two things happen here, one is that the function is sort of reflected because the coefficient is negative (i.e. positive quadratic = parabola concave upwards and negative quadratic = concave downwards) and two is that again as the absolute value of the coefficient increases, the function becomes skinnier.

Note: If the functions added/subtracted are of different degrees, then only the lower degree components have a small effect on the resulting function (i.e. the resulting function is simply shifted, with no bearing on its steepness/width).

2) Next, we look at what happens during multiplication and division of functions.

To better understand what happens to a functions graph, let's recall what happens when we multiply or divide functions.

f(x) = 2x
g(x) = 3x $^{2}$

f(x)g(x) = 6 $x^{3}$
g(x)/f(x) = $3/2x$

What do we notice here? One is that the leading coefficient changes, though we cannot generalise about its behaviour, we can say that how wide/skinny a function is is affected by multiplication/division of functions. Next, and more importantly is that the degree of the function changes. What this means is that for these functions and higher degree ones, the number of critical points is either increased or decreased in the process of multiplication and division. In other words, the bigger the degree of the function, the more critical points there are and vice versa. This is shown very evidently on its graph by how many times we see a plateau phase or a point on the domain where the tangent line to the curve has a slope of zero.

In general multiplying functions yields higher degree functions and thus more critical points and dividing yields lower degree functions with less critical points, but beware that there are many examples which do not follow this "rule of thumb" if you will.

Composition of Functions - Translate/Scale/Reflect Graphs

Link to video : http://www.youtube.com/profile?user=Math110Group13#grid/uploads

How to Translate, Scale and Reflect Graphs

There are three ways in which to change a graph, the most common terms known are: Translating, Stretching/Compressing, and Reflecting. These are the different changes in how a function could change, according to how the x or y of a function is changed.

Translating:

There are four different ways in which to translate a graph. Translating a graph simply means relocating or moving the graph while keeping its properties the same (eg. shape and size).

The first two ways in which to translate a graph is simply left and right.

1) To translate the graph left, we simply add a number to x.

For example : f(x) = x² → f(x) = (x+2)² - here we have added 2 to x

In this example, all the x-values of the x² function has shifted 2 units to the left. If the origin of the original function was at (0,0), then the new origin of the function would be (-2,0).

2) To translate the graph right, we simply subtract a number from x.

For example : f(x) = x² → f(x) = (x-2)² - here we have subtracted 2 from x

In this example, all the x-values of the function x² has shifted 2 units to the right. If the origin of the original function was at (0,0), then the new origin of the function would be (2,0).

Another way to translate a graph is translating it up or down.

3) To translate the graph up, we simply add a number to the whole function of f(x), not just x.

For example : f(x) = x² → f(x) = x²+2 - here we have added 2 to f(x)

In this example, all the y- values of the function x² has shifted 2 units up. If the origin of the original function was at (0,0), then the new origin of the function would be at (0,2).

4) To translate the graph down, we simply subtract a number to the whole function of f(x), not just x.

For example : f(x) = x² → f(x) = x²-2 - here we have subtracted 2 from f(x)

In this example, all the y-values of the function x² has shifted 2 units down. If the origin of the original function was at (0,0), then the new origin of the function would be at (0,-2).

Stretching/Compressing:

There are 2 different methods in which we can scale a graph, first we can compress a graph and second we can stretch a graph. Both methods can be done vertically and horizontally. In other words, we can have vertical compression, horizontal compression, vertical stretch, and horizontal stretch.

1) Vertical Stretching and Compressing

a) To stretch vertically, we multiply a factor of k to the entire function f(x). Think of this as pulling a rubber band upwards by a factor of k.

For example : f(x) = kf(x) ⇒ f(x) = x² → f(x) = kx²

All the y-values become increased by a factor of k. If we make a table of values and find that a point on the original graph of x² was (2,4), we would simply take the x-value of this point (2), and stick it in the new function where we would first square it 2 and then multiply it by k. This would enable us to find the new value of y for the new function. Given that k’s value is 3, the new values would be (2,12).

b) To compress vertically, we divide by a factor of k to the entire function f(x).

Think of this as pushing a rubber band into itself by a factor of k.

For example : f(x) = (f(x))/k ⇒ f(x) = x² → f(x) = (x²)/k

All the y-values become decreased by a factor of k. If we make a table of values and find that a point on the original graph of x² was (4,2), we would simply take the x-value of this point (4), and stick it in the new function where we would first square it by 2 and then divide it by k. This would enable us to find the new value of y for the new function. Given that k’s value is 2, the new values would be (4,8).

2) Horizontal Stretching and Compressing

a) To stretch horizontally, we divide a value of k to x only. NOT the entire function f(x).

For example : f(x) = f(x/k) ⇒ f(x) = x² → f(x) = (x/k)²

All the y-values change as the original values of x becomes divided by k. If we make a table of values and find that a point on the original graph of x² was (2, 3), we would simply take the x-value of this point (2), and stick it into the new function where we would first divide it to k. and then square it. This would enable us to find the new value of y for the new function. Given that k’s value is ½, the new point would be (2,1).

b) To compress horizontally, we multiply a value of k to x only. NOT the entire function f(x).

For example : f(x) = f(kx) ⇒ f(x) = x² → f(x) = (kx)²

All the y-values change as the original values of x becomes multiplied by k. If we make a table of values and find that a point on the original graph of x² was (3,7), we would simply take the x-value of this point (3), and stick it into the new function where we would first multiply it to k. and then square it. This would enable us to find the new value of y for the new function. Given that k’s value is 2, the new point would be (36,7).

Reflecting:

By reflecting a graph we are simply flipping it along the x-axis or the y-axis.

1) Reflecting along the y-axis

To reflect along the y-axis we place a negative in front of the x only. NOT the whole function f(x).

For example : f(x) = f(-x) ⇒ f(x) = x² → f(x) = (-x)²

To reflect the graph on the y-axis we simply take the whole graph and flip it along the y-axis, this means that all the values of x becomes the negative of itself while all the y-values stay the same. If we were to pick a point on a graph say, (9,9), in order to reflect it along the y-axis we simply take the negative of the original x-value so that the new point would be (-9,9).

2) Reflecting along the x-axis

To reflect along the x-axis we place a negative in front of the function f(x).

For example : f(x) = -f(x) ⇒ f(x) = x² → f(x) = -x²

To reflect the graph on the x-axis we simply take the whole graph and flip it along the x-axis this means that all the values of y becomes the negative of itself while all the x-values stay the same. If we were to pick a point on a graph say (8,5), in order to reflect it along the x-axis, we simply take the negative of the original y-value so that the new point would be (8,-5).