Factor both the numerator N(x) and the denominator D(x).

Cancel any common factors and simplify the function.

Equate the canceled factor to zero. This will give you the x-value of the hole.

Plug this x-value in the simplified function to find the y-value of the hole.

Plot that hole (or holes) on the graph.

VERTICAL ASYMPTOTES

Plot the remaining zeros of D(x) on the x-axis. Draw vertical dotted lines through them. These are the locations for the vertical asymptotes.

Find the sign of f(x), just before and after the dotted line. This you can do by finding the signs of the factors in the simplified function and resolving them. This will tell you if the graph is going asymptotic upwards or downwards near the dotted vertical line.

Find remaining zeros of N(x) on the x-axis. These are points where the graph crosses the x-axis.

HORIZONTALASYMPTOTES

Horizontal asymptotes occurs at either end of the graph as x goes to plus or minus infinity.

For n < m, the horizontal asymptote is y = 0 (the x-axis).

For n = m, the horizontal asymptote is y = a_{n} / b_{m}

For n = m+1, the asymptote is a slanted line, y = kx, found by dividing N(x) by D(x).

For n > m+1, there are no asymptotes; when n – m is even, both ends of the graph rise up when n – m is odd, the left end goes down while the right end rises up.

From the above data you can sketch a rough approximation of the shape of the graph.

The constant term of the polynomial shows the y-intercept of the graph.

Look at the highest degree of the polynomial. The corresponding graph will have one less bend. For example, if the degree of the polynomial is 3, the corresponding graph shall have two bends.

If all the zeros of the polynomial are real, then the corresponding graph will cross the x-axis as many times as the degree of the polynomial. For example, if the degree of the polynomial is 3, the corresponding graph shall cross the x-axis 3 times.

If the zeros of the polynomial are visible because the polynomial is factored; then, match the zeros to the values where the graph crosses the x-axis.

When two of the zeros are the same; then, the corresponding bend will simply touch the x-axis at that value.

When two of the zeros are imaginary; then, the corresponding bend will not cross or even touch the x-axis.

When all the zeros are real, the constant term of the polynomial shall be the product of the zeros.

When there are two graphs matching the polynomial, take the root that is not common in both the graphs, and plug it in the polynomial. You will know if that zero belongs to the polynomial or not.

This video explains how to read and write numbers up to trillions.

The NUMBERS are like words. They are made up of DIGITS, just like words are made up of letters.

In English, all the words are written with just twenty-six letters, from A to Z. In mathematics, all the numbers are written with just ten digits, from 0 to 9.

A DIGIT IS LIKE A LETTER. A NUMBER IS LIKE A WORD.

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Place Values

The Rule of Abacus gives us beads of values one, ten, hundred, etc. These values appear in numbers as PLACE VALUES as shown below. This number is made up of 3 hundreds, 9 tens, and 5 ones. It is read as “three hundred ninety-five.”

The place value of “one” may be thought of as a penny (one cent), the place value of “ten” may be thought of as a dime (10 cents), and the place value of “hundred” may be thought of as a dollar (100 cents).

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Written Numbers

A number written with one digit is called a single-digit number. There are nine single-digit numbers from 1 to 9. 0 (zero) is not a single-digit number because it represents nothing. It is called a place holder.

Example of a single-digit number: 7

A number written with two digits is called a double-digit number. There are ninety double-digit numbers from 10 to 99. You get ninety double-digit numbers by subtracting 9 from 99. You do not subtract 10 because it is included in double-digit numbers.

Example of a double-digit number: 43

A number written with three digits is called a three-digit number. There are nine hundred three-digit numbers from 100 to 999. You get nine hundred three-digit numbers by subtracting 99 from 999. You do not subtract 100 because it is included in three-digit numbers.

Example of a three-digit number: 478

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Group of Digits

The place values of “one-ten-hundred” make a group. A group contains numbers from 001 to 999. All three places in a group are shown by a digit even if a place has no value.

001 is the same number as 1

The most used groups are ONES, THOUSANDS, MILLIONS, BILLIONS, and TRILLIONS. These groups are arranged from right to left as shown below. The groups are separated by commas.

From right to left, the place values increase by a factor of ten. So, 10 “hundred” become 1 “thousand”; 10 “hundred thousand” become 1 “million”; 10 “hundred million” become 1 “billion”; and 10 “hundred billion” become 1 “trillion”. This pattern continues with higher place values.

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Reading & Writing Numbers

Here are some examples of reading and writing numbers.

We do not omit any place in a number. When a place has no count, we put “0” there as a place holder. For example, we write the number “302 trillion, 4 billion, 865 million, and Seven” as follows.

In this number the group “billion” has the value “004” (and not just 4). Since the group “thousand” is altogether missing, we put its value as “000”. The group “ones” has the value “007” (and not just 7).

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Check your Understanding

1. What are digits?

Digits are symbols used to write numbers.

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2. How many digits are there?

There are ten different digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

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3. How many real elephants are there in the room with you? What digit would you use to represent this number of elephants?

Most likely there is no real elephant in the room with you. You will use the digit ‘0’ in that case to represent the absence of elephants.

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4. Give examples for single-digit, double-digit, three-digit, and five-digit numbers.

7 is a single-digit number 32 is a double-digit number 483 is a three-digit number 63,153 is a five-digit number

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5. How many double-digit numbers are there?

From 10 to 99 (inclusive) there are 90 double-digit numbers.

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6. What are the three place values in a group (from right to left)?

One, ten and hundred

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7. What are the most used groups in a number (from right to left)?

Ones, Thousand, Million, Billion, and Trillion

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8. Write the number “Six billion, sixty-six thousand, sixty.”

6,000,066,060

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Final Thought

A DIGIT IS LIKE A LETTER. A NUMBER IS LIKE A WORD.

This video explains counting and the Rule of Abacus.

Mathematics starts with Arithmetic, and Arithmetic starts with counting.We learn to count on our fingers as follows.

We have two hands with a total of ten fingers.We can count with these fingers to find out how many things there are.

But to count beyond ten, it requires many hands. Alternatively, we can use an Abacus.

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The Abacus

An abacus is a counting board on which one can count to very large numbers.It has many wires. On each wire there are ten beads.

The count appears on the abacus when beads are moved to the right. The count on a wire is the number of beads on the right.

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The Rule of Abacus

“When all beads are counted to the right on a wire, they are replaced by counting one bead to the right on the next wire.”

Obviously, when there is no bead on the right, the count is zero.

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Counting beyond Ten

We count beyond 10 by moving beads on the first wire again.

Today, we may not use the abacus, but we still use the Rule of Abacus in our numbering system.

After 19, the next number is 20. After 29, the next number is 30.

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After 89, the next number is 90. After 99, the next number is 100.

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Computers and Binary Numbers

In our computers we use the numbers made up of 0’s and 1’s. These binary numbers may be created on an abacus that has only 2 beads on each wire. The Rule of Abacus applies to this “binary abacus” also.

The number of beads on a wire represent the “base” of the numbering system.

If you imagine an abacus with two beads on each wire, it will use the digits 0 and 1 only, because when two is counted the rule of abacus will apply and the number “two” will appear as “10”. The number “three” will appear as “11”. At “four” the rule of abacus will be applied twice. The number “four” will appear as “100”. You may construct all binary numbers this way.

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Check your Understanding

1. What is Counting?

The traditional way of counting consists of calling the first item as ONE, the next item as TWO, and so on. Counting can go on forever.

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2. What is the Rule of Abacus?

The RULE OF ABACUS is, “When all beads are counted to the right on a wire, they are replaced by counting one bead to the right on the next wire.”

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3. How is the Rule of Abacus helpful?

You need only as many digits to write the numbers as there are beads on a wire of abacus. One of those digits is always 0 (Zero). This is very useful when there are only two polarities to represent numbers as in the case of electronic computers.

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Final Thoughts

The very fact of counting makes mathematics fundamentally discrete. It cannot duplicate continuous reality, such as, the reality of PI (𝜋).