Math Homework Help

On the surface, math may seem less glamorous than biology or engineering, but the field, which deals with abstract concepts and equations, is integral to all other scientific fields.

As the United States faces a critical shortage of American scientists, the University of Wisconsin-Madison will lead an effort to attract more students to mathematics, ultimately providing the sciences with a stronger, smarter workforce.

We have four ways to alter the graph of a function: vertical translation, vertical scaling, horizontal translation, and horizontal scaling. Note that some or all four of these can be applied to a function at once in the sense that you may have to deal with a function of the form y = AsinB(x - C) + D. You should simply work these graphs out one step at a time, concerning yourself first with the affect of B and C, then the affect of the presence of A, and finally the affect of D.

Example:
Draw the graph of y =2 sin(x/2+?/6)+3

Solution
Let’s change this is of the form y = AsinB(x - C) + D
So y =2 sin(x/2+?/6)+3
can be written as y =2 sin ½ [x-(-?/3)]+3
Step 1: Draw sinx

B= ½ .so period is 4pi. That means it completes one cycle after 4pi.so stretch the graph horizontally so that its period is 4pi.

Next, we see that C=-pi/3. This will horizontally shift our graph pi/3units to the left. Thus, we now have

Now A=2, which will vertically scale our graph by a factor of 2. Thus, at this point, our range should now be [-2,2]

Finally, we must shift our graph vertically 3 units due to the presence of D=3. Hence, our range will now move to [1,5]

We can adopt the same method for drawing the graphs of other trigonometric functions. But remember first you should convert the given functions of the form

y = A sinB(x - C) + D

y = A cosB(x - C) + D

y = A tanB(x - C) + D

or accordingly, depending upon which of the trigonometric function is given.

Today I have a very simple puzzle for you.

It is a small town railway station and there are 20 stations on that line. At each part of the 2 stations the passengers can get tickets for any of the other 19 stations.

How many different kinds of tickets do you think the booking clerk has to keep?

Solution
The answer is simple. 19×20=380

The phase shift is the horizontal shift away from the standard graph of the trigonometric function.

In y = AsinB(x - C) + D,C is the phase shift.
For cosine function y = AcosB(x - C) + D and tangent function y = AtanB(x - C) + D also C is the phase shift.

If the phase shift is positive, there has been a horizontal shift to the right and if it is negative, there has been a horizontal shift to the left.

In reading off the phase shift, make sure you have the function in the form above.

For example, the phase shift of y = sin(2x - ? /2) is not ?/2. Rewrite the expression for the function in the required form to get
y = sin2(x - ?/4).
Now we see the correct phase shift, is ?/4.

In the graph we can see graph of sin(2x - ? /2) is shifted to ?/4 units to the right.

We know trigonometric functions are periodic functions. Sinx and cos x are periodic functions with period 2? or 360° . But tan and cot remain unchanged when x is increased by ? or 180° .
So, they are periodic functions with period ?.


The general form of the sine function is y = A sinB(x - C) + D
Here, the period is 2?/IBI.

The general form of the cosine function is y = A cosB(x - C) + D
We know cosine functions are identical to the sine functions. So, the period of cosine function is also 2?/IBI.
But for a tangent function. It is ? instead of 2? because the period of tan x is ?.
If the general form of a tangent function is y = A tanB(x - C)+ D,
its period is ?/IBI

There fore, the value of B is the key factor in determining the period of tangent functions. Change in its value changes horizontal stretching. When drawing the graph we have to “stretch” or ;“shrink” the graph horizontally by a factor of B.
Also, the period is unchanged by vertical scaling or shifting or by horizontal shifting.

I couldn’t post any articles last week. My daughter was not well. She is in her kinder garden classes. This year she already missed lots of classes due to various reasons. She will be five in next January. She is under aged when compared to other children. The average age of children in her class is above 5 ½ years. She is just above average in her studies. In math she is O.K. But she has to improve in language studies. I do not force her to study at all. If she likes, she study, otherwise not. Now I am thinking about repeating her in the same class next year also. But her teacher says there is no need to repeat her. So, I am in confusion.

Kathy and Jane were selling sweets in the market place. Kathy at 3 for a dollar and Jane at 2 for a dollar. One day both of them were obliged to return home when each had thirty sweets unsold. They put together the two lots of sweets and and handing them over to a friend, asked her to sell them at 5 for two dollars. According to their calculation, after all, 3 for 1 dollar and 2 for 1 dollar was exactly same as 5 for 2 dollars.

Now they were expecting to get 25 dollars for the sweets as they would have got, if sold separately. But much do their surprise they got only 24 dollars for the entire lot.

Now where did the 1 dollar go? Their friend was a cheat?

Solution
There isn’t really any mystery. While the two ways of selling are only identical, when the number of sweets sold at 3 for 1 dollar and 2 for 1 dollar is in proportion of 3:2. So , if Kathy had sold 36 sweets and Jane 24, they would have fetched 24 dollars (12 dollars each) , immaterial of, whatever sold separately or at 5 for 2 dollars. But if they had the same number of sweets which led to loss of 1 dollar when sold together, in every 60 sweets. So if they had 60 each (120 altogether) , there would be a loss of 2 dollars and so on.

In the case of 60, the missing 1 dollar arises from the fact that Kathy gains 2 dollars and Jane losses 3 dollars(If they share $12 each).

Kathy receives $9.5 and Jane $14.5, so that each loses $.50 in the transaction.

Periodic functions are functions that repeat its values over and over, after some definite period or cycle on a specific period. This can be expressed mathematically that A function f is said to be periodic if there exists a real T>0 such that f (x+T) = f(x) for all x.

The fundamental period of a function is the length of a smallest continuous portion of the domain over which the function completes a cycle. That is, it’s the smallest length of domain that if you took the function over that length and made an infinite number of copies of it, and laid them end to end, you would have the original function.

If a function is periodic, then the smallest t>0 ,if it exists such that f (x+t) = f(x) for all x, is called the fundamental period of the function.

The trigonometric functions sine and cosine are common periodic functions, with period 2?.
ie. sin (x+2?)= sin x , cos(x+2?)=cos x

But tan and cot remain unchanged when x is increased by pi.
ie. tan(x+?)=tan x, cot(x+?)= cot x
So, they are periodic functions with period ? .

An aperiodic function (non-periodic function) is one that has no such period

We know basic trigonometric functions are sinx, cosx, tanx.
These functions are periodic functions.( The period is the shortest interval over which the function runs through one complete cycle of its graph.)
Sinx and cos x are periodic functions with period 2?.
But tan and cot remain unchanged when x is increased by pi.So, they are periodic functions with period ?.

Amplitude

See the graph of sinx . We know its range is [-1, 1].

It is clear from the graph that its amplitude is 1

When we draw the graph of 2 sin x, we can see that its range is [-2, 2]
The multiplication factor 2 has “stretched” the graph of sinx vertically by a factor of 2, while retaining the same x-intercepts.

This vertical scaling factor is known as the amplitude of the function.

The amplitude of the sine and cosine functions is half the vertical distance between its minimum value and its maximum value.

For a function A sin x, its y values range from –A to +A
So amplitude is 1/2 of [A-(-A) ]=A

The vertical shifts do alter the greatest and least values that the function attains but do not alter the amplitude.

We can verify this by taking the examples 2sin x and 2sinx+3 For 2sinx,the minimum and maximum values are -2 and 2 .

Amplitude is ½ . 2-(-2)=2
For 2sinx +3, minimum and maximum values are 1 and 5 .

Amplitude is ½ (5-1)=2

In general , the amplitude of

y = A sinB(x - C) + D and
y = A cosB(x - C) + D ,where B is a non-zero real number, is IAI

The tangent function has no amplitude, because the tangent function has no minimum or maximum value.its range is (-infinity, infinity)

Today let’s relax a bit.

A peasant is convicted. He gets the death penalty. The judge allows him to say a last sentence in order to determine the way the penalty will be carried out. If the peasant lies, he will be hanged, if he speaks the truth he will be beheaded. The peasant speaks a last sentence and to everybody surprise some minutes later he is set free because the judge cannot determine his penalty.

What did the peasant said?

The peasant said: “I shall be hanged!”

If the peasant was lying, he would be hanged. But that’s what the peasant was saying. So he speaks the truth. But if he speaks the truth, he would be beheaded, so then he was not speaking the truth. So it is impossible for the judge to determine whether the peasant speaks the truth or not. So therefore the judge cannot determine the penalty and sets the peasant free.

Don’t go away……. I need your help…….

My 2 year old son put a coin in an empty bottle and insert a cork into the neck of the bottle.
The bottle has a very nice shape. So I don’t want to break it.
How could I remove the coin without taking the cork out ?

Answer

Don’t worry for a solution. He himself found a way and took it out. He pulled the cork into the bottle and the coin came out.