Function Notation Graph

Introduction:

Function in math describes a quantity of expression, when an n input is given to the expression we get an output called a function. It assigns a unique value to the specific expression. The function notation consists of even functions and odd functions. Here we will see the graph of the function notation in detail.

If f(x) = F (-x) then the function is even function.

Example: f(x) = cos x is an even function.

If f(x) ≠ f (-x) then the function is odd function.

Example: f(x) = sin x is odd function.

Below is the example for Function Notation Graph:

Function notation Graph - Example:

Graph the function y = x² – x

Solution:

Step 1: Find the values in the table.

x	-2	-1	0	1	2	3
y	6	2	0	0	2	6

Step 2: Design these points using function notation graph.

Step 3: Since y = 0 for both x = 0 and x = 1,

Step 4: The above function notation graph clear that the arc is continuous.

Hope you like the above explanation, Please leave your comments, if you have any doubts.

Methods of factoring trinomials calculator

Introduction:

The equation or a function is in the structure of ax²+bx+c =0 (where a≠0, b, c are constants ) called as trinomials.We can also refer it as a quadratic function. An algebraic expressions which has 3 terms known as trinomials.The trinomials having highest power 2.The trinomials have the two roots. There are two ways to factor the trinomial according to the co-efficient of x² . In the following section we are going to learn how to factor the trinomials by using trinomials calculator.

Factor Trinomials Calculator Method 1:

For factorind the trinomials we must know the below two methods.

Factor trinomial calculator Method 1:

If the coefficient of x² is one. That is a=1.

x²+bx+c=(x-r₁)(x-r₂), In this r₁ and r₂ are the roots of the trinomial equation

(x - r₁) and (x - r₂) are the factors of the trinomial.

Factor Trinomials Calculator Method 2:

In the trinomial equation ,the coefficient of x² has any value except 1. That is a≠1.

Hope you like the above explanation. Please leave your comments, if you have any doubts.

About General Circle Equation

Introduction to general circle equation:-

A circle is a simple shape of Euclidean geometry consists of that point in plane which is equidistant from a given point called the center. The common distance of the point of a circle from its center is called its radius. Circle simple close curve which divide the plane into two regions, an interior and an exterior. The circumference of a circle is the border of the circle (especially when referring to its length). In this article we shall discuss about general equation of circle.

General Form Circle Equation Example Problem:-

The circle equation of center radius of the circle in the form of (x – h) 2 + (y – k) 2 = r2. The center of the circle is starting at the point of (h, k) and the radius of the circle starting "r". This format of the circle equation is helpful, for easily find the center and the radius of the circle.

Example 1:-

Find the general equation of a circle, the center is at (2, - 6) and radius 6

Solution:-

Given (h, k) = (2, - 6) and r = 6

Substitute h, k vale and r value in the standard equation

(x - 2)² + (y - (- 6)) ² = 6²

(x - 2)² + (y + 6)² = 36

Different types of Variables

Introduction:

Quantities such as height, weight, age, amount can have several different values. Quantities which can assume different numerical values are called variables. A variable is any alphabet or combination of alphabets that contain the constant value. It can be vary. For example the speed of the vehicles is a variable. Variables with similar roles are mentioned with consecutive alphabets.

Variables are of two types:

(a) Continuous

(b) Discrete.

Continuous Variable

Consider an example. A person was asked to measure the thickness of a coin. He recorded the following readings:

(i) 0.2 cm with ruler

(ii) 0.23 cm with vernier

(iii) 0.231 cm with micrometer

The accuracy of thickness of the coin depended on the instrument used for measuring the thickness. Thus the thickness of the coin is a continuous variable. Variables like height, weight, time are also called continuous variables.

Discrete variable

Consider an example: The number of pupils in a class can be 30, 34, 40. But it cannot be 30.2, 35.1, because number of pupils can be expressed only as a whole number. Such numerical as number of children, trees, sheep are called discrete variables.

How to solve radicals equations

Introduction:

An equation with radical is an equation of the form X^m/n = a, for m, n integers.

Steps for solving equations with radicals:

When a variable is inside a square root, first take one of the radicals to one side of the equation. Squaring on both sides of the equation, we can eliminate the radicals (square root).

Now, depending on the equation, the variable involved in the new equation, either linear or quadratic methods used for solving the equation for unknown value. Now, we are going to see some radical equations problems.

How to solve radicals equations:

Example radical equation:

Find the value of x by solving the equation: `sqrt(x)` +8=9

Solution:

Subtract 8 on both sides

`sqrt(x)` +8-8=9-8

`sqrt(x)` =1

To eliminate square root, take square on both sides of the equation, we get

x=1²

x=1

So, the answer is x=1.

Example of solving Linear Equations in Two Variable

Introduction:

Solving linear equations in two variables is a process of finding the value of the unknown quantity for which the equation is true. The value so found is called the root or solution of the equation. The process of finding the value of the unknown quantity for which the equation is true, is called solving the equation. The value so found is called the root or solution of the equation.

An equation whose graph is a straight line is called a linear equation. (linear means straight). An equation of degree one is linear equation in one variable and with two variables is called as linear equation in two variable.

Example of solving Linear Equations in Two Variable

Example 1:

x = 1, y = 1 is a solution of 2x + 3y = 5

2(1) + 3(1) = 5

x = -2, y = 3 is also a solution of 2x + 3y = 5

2(-2) + 3(3) = 5

-4 + 9 = 5

Similarly, we can find many more solutions for 2x + 3y = 5

Example 2:

Find four solutions of the equation 2x + y = 5.

2x + y = 5

y = 5 - 2x

Put x = 0, y = 5

x = 1, y = 5 - 2 = 3

x = 2, y = 5 - 4 = 1

x = 3, y = 5 - 6 = -1

Hope you like the above explanation, Please leave your comments, if you have any doubts.

Ratios

Introduction:

This chapter takes us much further and adds to our already existing knowledge about Ratio and Proportion. We will have to use many of the algebraic manipulations learnt earlier, use them with expertise to solve problems in this chapter.

Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20. Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the numbers would have been "20 to 15". Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio: odds notation: 15 : 20 fractional notation: 15/20 You should be able to recognize all three notations; you will probably be expected to know them for your test. Given a pair of numbers, you should be able to write down the ratios. For example: There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats. Consider the above park. Express the ratio of geese to ducks in all three formats.

Algebraic Equation

Basics of the Equation

The diagram on the right shows a basic equation. This equation is similar to problems which you may have done in ordinary mathematics such as:

__ + 16 = 30

You could easily guess that __ equals 14 or do 30 - 16 to find that __ equals 14.

In this problem __ stood for an unknown number; in an equation we use variables, or any letter in the alphabet.

When written algebraically the problem would be:
x + 16 = 30

and the answer should be written:
x = 14

Solving Equations

These equations can be solved relatively easy and without any formal method. But, as you use equations to solve more complex problems, you will want an easier way to solve them.

Pretend you have a scale like the one shown. On the right side there are 45 pennies and on the left side are 23 pennies and an unknown amount of pennies. The scale is balanced, therefore, we know that there must be an equal amount of weight on each side.

As long as the same operation (addition, subtraction, multiplication, etc.) is done to both sides of the scale, it will remain balanced. To find the unknown amount of pennies of the left side, remove 23 pennies from each side of the scale. This action keeps the scale balanced and isolates the unknown amount. Since the weight(amount of pennies) on both sides of the scale are still equal and the unknown amount is alone, we now know that the unknown amount of pennies on the left side is the same as the remaining amount (22 pennies) on the right side.

Hope you like the above explanation, Please leave your comments, if you have any doubts.

Rounding Numbers

Rounding Numbers:

When you have to round a number, you are usually told how to round it. It's simplest when you're told how many "places" to round to, but you should also know how to round to a named "place", such as "to the nearest thousand" or "to the ten-thousandths place". You may also need to know how to round to a certain number of significant digits; we'll get to that later.

In general, you round to a given place by looking at the digit one place to the right of the "target" place. If the digit is a five or greater, you round the target digit up by one. Otherwise, you leave the target as it is. Then you replace any digits to the right with zeroes (if they are to the left of the decimal point) or else you delete the digits (if they are past the decimal point).

I'll use the first few digits of the decimal expansion of pi: 3.14159265... in the example below.

• Round pi to five places.

"To five places" means "to five decimal places". First, I count out the five decimal places, and then I look at the sixth place:

3.14159 | 265...

I've drawn a little line separating the fifth place from the sixth place. This can be a handy way of "keeping your place", especially if you are dealing with lots of digits.

The fifth place has a 9 in it. Looking at the sixth place, I see that it has a 2 in it. Since 2 is less than five, I won't round the 9 up; that is, I'll leave the 9 as it is. In addition, I will delete the digits after the 9. Then pi, rounded to five places, is:

= 3.14159




Hope you like the above Explanation, Please leave your comments, if you have any doubts.

Solving Fractions

Definition:

We all have learnt fractions and decimals in earlier classes. The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. We also studied comparison of fractions, equivalent fractions, representation of fractions on the number line and ordering of fractions.

Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction. We shall now learn multiplication and division of fractions as well as of decimals

How to Do Fractions:

1. In order to do fractions we first need to find the common factor of both the numerator and denominator (A common factor is any number that divides both the numerator and denominator). Like 3 divides 6 and 21.
2. Then continue dividing the fraction with the common factor till there are no more common factors in the numerator and denominator.
3. Now the fraction is called as simplified fraction as no common factors remaining which can divide both the numerator and denominator.

Example of simplifying fractions:

Lets take 12/14. Here both 12 and 14 are divisible by 2 so lets divide the fraction with 2.
So we get 12/14 = 6/7 Now 6/7 does not have any common factors so 6/7 is the simplified fraction
So both 18 and 42 are divisible with 2,3,6

Place Value

Definition:


The The idea of place value is at the heart of our number system. First, however, a symbol for nothing--our zero--had to be invented. Zero "holds the place" for a particular value, when no other digit goes in that position. For example, the number "100" in words means one hundred, no tens, and no ones. Without a symbol for nothing, our decimal number system wouldn't work.

Beginning with the ones place at the right, each place value is multiplied by increasing powers of 10. For example, the value of the first place on the right is "one", the value of the place to the left of it is "ten," which is 10 times 1. The place to the left of the tens place is hundreds, which is 10 times 10, and so forth.
For easier readability, commas are used to separate each group of three digits, which is called a period. When a number is written in this form, it is said to be in "standard form."

Examples
Numbers can be represented in many ways, but standard form is usually the easiest and shortest way. Here are some numbers expressed in different forms, with their standard form shown alongside. Which form do you think is the best?
Example 1
one billion, sixty million, five hundred twenty thousand
1,060,520,000
Example 2
four hundred sixteen thousand, seven hundred thirty-one
416,731
Example 3
6,000,000 + 70,000 + 20 + 1
6,070,021