Algebra 2 Help and Answers

Algebra 2 help us find the unknown quantities with the help of known quantities. In algebra, we frequently use letters to represent numbers. Algebra 2 help includes real numbers, complex numbers, vectors, matrices etc. Algebra 2 is the study of the rules of relations and operations, and the constructions arising from them. An algebraic expression represents a scale that gets added or subtracted or multiplied or divided on both sides. In algebra 2 help and answers numbers are considered as constants.


Algebra 2 help and answers example Questions:


The following problems gives different answers to algebra 2 problems.

Ex 1:  Determine all real solutions to the equation

Sqrt (2 x + 13) = x - 5

Sol :   Given
sqrt (2 x + 13) = x - 5

We raise both sides to power 2.
[Sqrt (2 x + 13)] 2 = (x - 5) 2

And simplify.
2x +13 = x 2 - 10 x +25

Write the equation with right side equation to 0.
X 2 - 8 x + 12 = 0

It is a quadratic equation with 2 solutions
x = 6 and x = 2

Ex 2 :  Determine all real solutions to the equation

Sqrt (x + 8) = 12

Sol :  Given
sqrt (x + 8) = 12

We raise both sides to power two(2) in order to clear the square root.
[Sqrt (x + 8)] 2 = 12 2

And simplify
x + 8 = 144

Solve for x.
x = 136

Ex 3:   Determine all real solutions to the equation

Sqrt (x 2 – 13x+72) = 6

Sol :   Given
Sqrt (x 2 – 13x+72) = 6

We raise both sides to power 2.
[Sqrt (x 2 – 13x+72)] 2 = (6) 2

And simplify.
X 2 – 13x+72= 36

Write the equation with right side equation to 0.
X 2 - 13 x + 36 = 0

It is a quadratic equation with 2 solutions
x = 9 and  x = 4


Algebra 2 help and answers practice problems:


1) Determine all real solutions to the equation

Sqrt (x + 25) = 13

Ans : x=12

2) Determine all real solutions to the equation

Sqrt (x 2 – 11x+55) = 5

Ans : x=5 and x=6

3) Determine all real solutions to the equation

Sqrt (2 x + 24) = x

Ans : x=6 and x= - 4

Complex Polygon

Complex polygon is a polygon whose sides cross over eachother one or more times. If the number of cross over increases, the complexity of the polygon also increases. Below are some of the complex polygons

Area of a Complex Polygon


There are three steps to find the area of the complex polygon. They are as follows:

Step 1: Break the polygon into simple rectangle, square or triangle.

Step 2: Find the area of all rectangle, square or triangle.

Step 3: Add all the area and get the area of complex polygon.

With the help of the above steps, we can easily find the area of any complex polygons.


Examples

Given below are some of the examples to find the area of a complex polygon.

Example 1:

Find the area of following polygon


DE= 5cm, EF =8cm, FD = 5 cm, AB = 14cm, CD = 10cm and the distance between these two parallel side AB and CD is 5cm

Solution:

Step 1: Break the polygon into triangle and trapezoid.


Here, the given complex polygon is braked into two simple polygon. It is separated at the point D

Step 2: Find the area of triangle (D1EF) and trapezoid (ABCD2).

Area of triangle (D1EF)

Value of s = (D1E + EF + FD1) / 2

Value of s = (5+8+5) / 2

Value of s = 9 cm

Area of triangle (D1EF) = SquareRoot (s(s - D1E)(s - EF)(s - FD1))

= SquareRoot (9(9 - 5)(9 - 8)(9 - 5))

= SquareRoot (9 x 4 x1 x 4)

=  SquareRoot (144)

= 12 square cm

Area of Trapezoid (ABCD2) =  ½ ( AB +CD2 ) x ( Perpendicular distance between AB  and  CD2 )

= ½ (10 +14) x 5

= ½ (24) x 5

= 12 x 5

= 60 square cm

Step 3: Add all the area of triangle and trapezoid to get area of complex polygon.

Area of Complex polygon = Area of triangle (D1EF) + Area of Trapezoid ABCD2)

= 12 square cm  +  60 square cm

= 72 square cm

Binomial probability formula

Binomial is deals with the polynomial that has two terms is known as a binomial e.g.5x + 3y, 2x^2 – 5xy.Probability is the arithmetical quantify of the possibility of an event to occur. If in an examine there are n feasible ways completely and mutually exclusive and out of them in m ways in the event A occur, it is given by (m / n) if in a random sequence of n trails of an events, M are favor to the event, then ratio is (M / n).by studying in both combined as binomial probability as P(A) = m / n.

The possibility of an event can be conveyed as a binomial probability if its conclusions can be wrecked down into two probability of p and q, where p and q are balancing (i.e. p + q = 1)


Binomial Probability Formula - Explained


The probability of getting exact value of k in n trials is given by using the formula,

`P(X = x) = ((n),(k)) p^k q^(n-k) `
where
` q= (1-p)`
or
`P(X = r) =nCr p^r(1-p)^(n-r)`

where
n = Number of events.
r = Number of successful events in the trials.
p = Probability of success in the single trial of the events.
nCr = `(( (n!) / (n-r)! ) / (r!))`
1-p = Probability of failure in the trial.


Examples on Binomial Probability Distribution

Example 1:

Assume that there are taking about 10 question multiple choice test. If each question has four choices and you have to guess on each question, what is the probability of getting exactly 7 questions correct using binomial Probability formula?


Solution:
General Formula for finding solution is,

`P(X = x) = ((n),(k)) p^k q^(n-k) `

n = 10
k = 7
n – k = 3
p = 0.25 = probability of guess the correct answer on a question
q = 0.75 = probability of guess the wrong answer on a question

P(7 Correct guesses out of 10 questions) = `((10),(7))` (0.25)7(0.75)3
? 0.0031 approximate value

Therefore if someone guesses 10 answers on a multiple choice test with 4 options, they have about a 5.8% chance of getting 5 and only 5 correct answers. If 5 or more correct answers are needed to pass then probability of passing can be calculated by adding the probability of getting 5 (and only 5) answers correct, 6 (and only 6) answers correct, and so on up to 10 answers correct. Total probability of  5 or more correct answer is approximate percentage is  7.8


Example 2:
Suppose a die is tossed 7 times. What is the probability of getting exactly 3 fours?

Solution:

General Formula for finding solution

`P(X = r) =nCr p^r(1-p)^(n-r)`

Step 1:
Number of trials n = 7
Number of success r =3

Probability of success in any single trial p is given as 1/6 or 0.167

Step 2:

To calculate nCr formula is used.

nCr = `(n!)/((n-r)!(r!))`

= `(7!)/((7-3)!(3!))`

= `(7!)/((4)!(3!))`

= `(5040)/((24)(6))`

= `(5040)/(144)`

= 35

Step 3:

Find pr.
pr =  0.1673
= 0.004657463

step 4:

To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.167 = 0.833
n-r = 7-3 = 4

Step 5:

Find (1-p)n-r.
= 0.8334 = 0.481481944

Step 6:

Solve P(X = r) = nCr p r (1-p)n-r
= 10 × 0.004657463 × 0.481481944
=  0.0224248434

The probability of getting exactly 3 fours is 0.0224248434

Solving Online Venn Diagram of Quadrilaterals

We know that on combining three non-collinear points in pairs, the figure obtained as a triangle. Now, let us mark four points and see what we obtain on combining them in pairs in some order. Such a figure created by combining four points in an order is called quadrilateral. In a quadrilateral, a pair of opposite sides is parallel.

In online, students can learn about quadrilaterals. Through online, students can have interactive sessions with tutors. Online is one of the efficient tools for one to one learning. In this article, we are going to discuss about solving online venn diagram of quadrilaterals.

Solving Online Venn Diagram of Quadrilaterals - Definition and its Types:

Definition of Quadrilateral:

Quadrilateral is a geometrical figure that consists of four ends called verticals joined to each other by straight-line segments or sides called edges. The quadrilateral are classified depends on the length of side, angle and diagonals of object.

Solving online venn diagram of quadrilaterals - Types:

There are two types of Quadrilaterals in geometry. The following are the two types of Quadrilaterals.

Convex quadrilaterals:

Parallelograms
Square
Rectangle
Rhombus

Concave quadrilaterals:

Trapezium
Kite

Solving online venn diagram of quadrilaterals:



Solving Online Activities with Quadrilaterals - Descriptions and Example Problems:

Parallelogram:

Parallelogram is one of the classifications of quadrilateral
Parallelogram consists of parallel opposite sides
Parallelogram consists of equal length
Parallelogram consists of equal opposite angles


Formula:

Area = b × h sq. units (b – Base, h - Height)

Example:

The base of a parallelogram is 19 and the height is 12. What is the area of this parallelogram?

Solution:

Area of parallelogram = b * h

= 19 * 12

= 228sq.units

Square:

Parallel lines are lines are exactly the same length and height that has never cross.

Square is a one of the classifications of quadrilateral
A square shape is same to both rhombus and rectangle
Square consists of equal parallel sides and also all angles are right angle (90°) only
The Square is a parallelogram having an angle, equal to right angle and adjacent sides equal.


Formula:

Area = a2    (a - area)

Example:

Find the area of square, when side length is 22cm

Solution:

Area of square = a2

= 22 * 22

= 484 cm2

Rhombus properties:

Rhombus is a parallelogram having its adjacent sides equal but more of whose angles is a right angles.

Rhombus consists of four-sided geometrical shape and also all sides have same length
Rhombus consists of parallel opposite sides
It consists of equal opposite angles
It is also one of the parallelograms


Formula:

Area = `1 / 2` (d1 × d2) sq. units

Where, d1, d2 are diagonals.

Example:

Find the area of rhombus d1 = 20 cm, d2 = 29 cm

Solution:

Area of rhombus = `1 / 2` (d1 * d2)

= `1 / 2` (20 * 29)

= (10 * 29)

= 290 cm2

Trapezium:

A quadrilateral should have at least one pair of parallel sides. The quadrilateral one side of opposite sides are parallel if non parallel of opposite sides of a trapezoid are congruent it is called as isosceles triangle.

Trapezium is also otherwise known as trapezoid
Trapezium is one of the main types of quadrilateral
Trapezium has one pair of parallel that present in opposite sides


Formula:

Area = `1 / 2` (a + b) × h sq. units.

Example:

Find the area of trapezoid height = 11 cm, a = 20 cm, b = 29 cm

Solution:

Area of trapezoid = `1 / 2 ` (a + b)* h

=` 1 / 2` (20 + 29) * 11

= `1 / 2 ` (49) * 11

= 24.5 * 11

= 269.5 square cm

Right Triangle Similarity Theorem

The right triangle similarity theorem is otherwise called as Pythagoras theorem. This right angle theorem was introduced by Philosopher and Greek Mathematician, Pythagoras. The right triangle is a part of geometrical figures. This theorem is used for finding the length of any sides of a right triangle. The right triangles are special triangles that contain only one right angle. Here the right triangle is a triangle that measuring an angle `90^o`

Statement for Right Triangle Theorem:

In terms of area, we can define this theorem. In any right triangle, the area of the square whose side is hypotenuse is equal to the sum of the areas of the other two sides. Here the hypotenuse refers to the side that opposite to the right triangle. The other two sides of a triangle meet at right angle.



Here Hypotenuse is the side opposite to the right angle. an adjacent side is the side adjacent to the given angle and the opposite side is the side that opposite to given  .

According to this theorem, the equation can be given as

(Opposite)2+ (Adjacent)2 = (Hypotenuse)2

Here, an angle  value lies between 0 and 90 degree. Here, 90 degree is one of the interior angle and the other two interior angles are complementary. Complementary means the angle value less than 90 degree. The other two interior angle sum should be equal to 90 degree.  This right triangle similarity theorem can also be used in trigonometric functions. This right triangle forms six possible ratios in trigonometry.



Other Forms of the Equation:



Here consider a is opposite side, b is an adjacent side and c is hypotenuse. Therefore, an equation can be written as a2 + b2 = c2. To find c, an equation can be written as c= `sqrt(a^(2)+b^(2))` . If c is known value, the length of one side is given; the following equations can be used;

a =`sqrt(c^(2)+b^(2))`   (or)       b = `sqrt(c^(2)-c^(2))`

Proof:

This theorem proof is based on the proportionality of two similar triangles. It depends on the ratio of any two corresponding sides of similar triangles.



Let ABC represent a right triangle, with the right angle located at B. Here, H is the altitude drawn from B and it also intersects AC. The point H divides the hypotenuse c into two parts d and e. The new triangle ABH is similar to the triangle ABC because both the triangles have a right angle. Similarly, the triangle BCH is also similar to the triangle ABC. Thus the proof of similarity of the right triangles requires the Triangle postulate i.e., the sum of the angles in a triangle is 2 right angles, and is equivalent to the parallel postulate. Similarity of the right angle triangles leads to the equality of ratios of their corresponding sides:

a/c =e/a and b/c= d/b

In this the first result is equal to cosine angle and the second result is equal to sine angle.

These ratios can be written as:

a2= c*e and b2=c*d

Summing these two equations we get:

a2 + b2 = (c*e) + (c*d) = c* (e + d) = c*c = c2

Therefore a2+ b2 = c2

Hence, right triangle similarity theorem is proved.

Graphing Calculator Polar Coordinates

In this article, we are going to study about graphing calculator polar coordinates. It is very simple to work on the polar coordinates graphing calculator. For example, if we want to graphing radians r = 3 and `theta=45^o` on the polar graph paper, we have to just enter `3` in the radians text box and `45^o` in angles in degrees text box and then click the enter button, Now the calculator automatically plot the polar coordinates on the display of the polar graph paper. These process are shown in the below figure.




Graphing Calculator Polar Coordinates - Example 1:

Study and graphing of the given polar coordinates.

(2, 50o), (3, 210o), and (-2, 135o).

Solution:

Graphing the given polar coordinates are on the below polar graph paper:



Steps for plotting polar coordinates (2, 50o):

1:   The radian 2 is marked from the origin (pole) on the positive side of the x axis.

2:   The given angle 50o is rotated in anticlockwise directions and then plot the required polar coordinates (2, 50o) as shown in figure.

Steps for plotting polar coordinates (3, 210o):

1:   The radian 3 is marked from the origin (pole) on the positive side of the x axis.

2:   The given angle 210o is rotated in counterclockwise directions and then plot the polar coordinate (3, 210o) as shown in figure.

Steps for plotting negative polar coordinates (-2, 135o):

1:   The radian 2 is marked from the origin on the positive side of the x axis.

2:   The given angle 135o , is rotated in anticlockwise directions and it is determined.

3:   Now we can extend the 135o degree line in opposite direction (315o) where we get  the radian 2 in negative sign and then plot the polar coordinates (-2, 135o).

Graphing Calculator Polar Coordinates - Example 2:

Study and graphing of the given polar coordinates by using the above calculator.

(2.5, 60o), (-1, 75o), and (3, -30o).

Solution:
Polar coordinates are on the below polar graph paper:



Steps for plotting polar coordinates (2.5, 60o):

Step 1:   The radian 2.5 is marked from the origin (pole) on the positive side of the x axis.

Step 2:   The given angle 60o is rotated in anticlockwise directions and then plot the required polar coordinates (2,.5 60o) as shown in figure.

Steps for plotting polar coordinates (3, -30o):

Step 1:   The radian 3 is marked from the origin (pole) on the positive side of the x axis.

Step 2:   The given angle 30o is rotated in clockwise directions and then plot the polar coordinate (3, -30o) as shown in figure.

Steps for plotting negative polar coordinates (-1, 75o):

Step 1:   The radian 1 is marked from the origin on the positive side of the x axis.

Step 2:   The given angle 75o , is rotated in anticlockwise directions and it is determined.

Step 3:   Now we can extend the 75o degree line in opposite direction (255o) where we get  the radian 1 in negative sign and then plot the polar coordinates (-1, 75o).

Definition Non Linear

In math, a nonlinear system is one of the system which is not linear, that is, a system which does not assure the superposition principle, or whose result is not directly proportional to its input. The equation used to solve the non linear problem is f(x)=C. This function is called as linear if f(x) is a non linear. This article gives some of the examples about the non linear by using this definition.


Examples of Definition Non Linear:

By using the definition of non linear solve the following examples.

Example problem1:

Use the following function rule to find f(7).

`f(x)=2x^2`

Solution:

Given `f(x)=2x^2` ; Find f(7).

Plug x = 7 into the function and simplify.

`f(x) = 2x^2`

`f(7) = 2(7)^2`            Plug in x = 7

f(7) = 2(49)                  Square

f(7) = 98                       Multiply

Which is the required solution.

Example problem2:

Use the following function rule to find f(8).

`f(x) = (x - 4)^2`

Solution:

Given:

`f(x) = (x - 4)^2`

Find: f(8)

Plug x = 8 into the function and simplify.

`f(x) =(x - 4)^2`

`f(8) = ( 8 - 4)^2`            Plug in x = 8

`f(8) = (4)^2`                  Subtract

f(8) = 16                          Square

Which is the required solution.

Example problem3:

Use the following function rule to find f(1).

f(x) = 5 – 4|x|

Solution:

Given: f(x) = 5 – 4|x|; Find f(1)

Plug x = 1 into the function and simplify.

f(x) = 5 –4|x|

f(1) =5 – 4|1|                  Plug in x = 1

f(1) =5 – 4(1)                 Take the absolute value

f(1) =5 – 4                       Multiply

f(1) = 1                            Subtract

Which is the required solution.

Practice Problems of Definition Non Linear:

Problem 1:

Use the following function rule to find f(7).

f(x) = -9|x| + 4

Solution:

f(7) = -59

Problem 2:

Use the following function rule to find f(-8).

`f(x) = 3x^2`

Solution:

f(-8) = 192

Relevance and Application of Exponential Functions in Real Life Situations

In this article relevance and application of exponential functions in real life situations , we will see how exponential function used in real life. Exponential function is used in more real life application like compound interest, problem based on population, problem based on radioactive decay, mortgage problems. Let us work out some problem to make understand that application of an exponential function.

Exponential Function Growth:

Exponential function growth`g=c(p)^t where,`

c-Number at initial

p-growth factor (p =1+r,here r is given growth percentage)

t-time

Example 1 -  Relevance and application of exponential functions in real life situations

A group of 1000 people increase by 5% in an hour near to accident place. How many people will be in the crowd after 3 hour?

Given:

c=1000

p =1+r =1+0.05=1.05

t =3 hour

Solution:

Exponential function growth g=c(p)^t

Substitute the given data in the formulae

g=1000(1.05)^3

g=1000(1.1576)

g=1157.6

g=1158 people

Exponential Function Decay:

Exponential function decay `d=c(p)^t`

where,

c-Number at initial

p-growth factor

(p=1-r,here r is given decay percentage)

t-time

Example 2: Relevance and application of exponential functions in real life situations

The price of a violin is  $1,000 which decreases at a rate of interest of 3%.What is the price of a violin after 2 year?

Given:

c=1,000

p=1-r=1-0.03=0.97

t=2 year

Solution:

Exponential function decay `d=c(p)^t`

`d=1,000(0.97)^2`

d=1,000(0.9409)

d=940.9

d=941

The price of a violin after 2 year=$941

Example 3: Relevance and application of exponential functions in real life situations

John invests $50,000 for 2 year with the interest of 4% compounded half yearly .Find out the compound interest for his investment at a given rate of interest?

Given:

P=$50,000

R=4%

N=2 year

Solution:

Compound interest =` P[1+(R/2)/100]^(2n)`

`=50,000[1+(4/2)/(100)]4`

`=50,000[1+(2/100)]4`

`=50,000xx(102/100)xx(102/100) xx(102/100) xx(102/100)`

=$50,000(1.0824)

=$54121.6

Compound interest=Total amount-principle

=54,121.6-50,000

=$4,121.6

Example 4: Relevance and application of exponential functions in real life situations

Calculate how long blood clotting cell will take to produce 20,400

Solution:

We need to consider  assume function to calculate population based on problems

`f(t)=2^t`

`20,400=2^t`

Find the natural logarithm on both side

`ln(20,400)=ln(2^t)`

ln(20,400)= t ln(2)

`t= ln(20,400)/ ln(2)`

`t=(9.923)/0.693=14.29`

After 14.32 min the blood clotting cell can produce 20,400  cell.

Angle Addition Property

In this article we are going to discuss Angle Addition Property , the terms related to Angle Addition Property and some solved problems on Angle Addition Property .

Introduction to angle addition property :

The angle addition postulate states that if a point is within an angle and you add the two angles that are made by drawing a line through the point that the total will equal the large angle. ...




Angle Addition Property or postulate says that if there is a line segment SV lies in the interior of the angle TSR then Angle TSV + Angle VSR = Angle TSR

For example if Angle TSR = 40 degree and Angle TSV = 15 degree then the other angle will be of 25 degree.

The same property can be applied if there are two or more lines lies in the interior of an angle.

There are some important terms that are associated with Angle Addition Property

If two or more angle sums to 90 degree they are called complementary angles.



If two or more angles are lies in a straight line then definitely sums to 180 degree and called Supplementary Angle.



Two or more angles sharing same side are called Adjacent Angles.



Having problem with ------ Read my upcoming post, i will try to help you.

Lets Learn more about Angle Addition Property in Supplementary Angles



Supplementary Angles

What is the measure of Angle B if angle A is 120 degree?

Angle B + Angle A = 180

Angle B + 120 = 180

Angle B = 60 degree

Problem Based on Angle Addition Property.

Lets learn more about Angle Addition Property



Suppose Angle CAD is a complementary angle
‹CAB = 40 degree

‹EAD = 40 degree

‹BAE =  ?

Solution Angle CAB + Angle EAD + Angle BAE = Angle CAD
40 degree +40 degree + Angle BAE = 90
Angle BAE = 10 degree
What if I remove the line AB
calculate ‹CAE

Solution

‹CAE = 50 degree (‹CAB + ‹ BAE)

What if the three of the angle are of equal

Solution :Suppose each angle = x degree
X+x+x = 90
3x= 90
x= 30 degree
If Angle EAD is thrice to angle CAB

Angle BAE is 10 more to Angle CAB

Measure each angle

Let Angle CAB = x degree
Angle BAE = x+10 degree
Angle EAD = 3x
x+x+10+3x = 90
5x+10= 90
5x=80
x=16
Therefore Angle CAB ,BAE ,EAD are 16 ,26 ,48 degree respectively.

Determine the Slope and Y Intercept

Slope:

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is




Y intercept:

In the coordinate system, the y intercept of a line is a point at which the line cuts the Y-axis. The y-intercept of a line is denoted as (0, y)




Example Problems to Determine the Slope and Y Intercept:

Example problem 1:

Determine the slope and y intercept of y = 7x - 12

Solution:

Step 1: Given equation

y = 7x - 12

Step 2: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 3: Compare the given equation with the slope intercept form

By comparing the given equation with the slope intercept form, we get

m = 7     and b = - 12

Step 4: Solution

Therefore,  Slope = 7

y intercept = - 12

Example problem 2:

Determine the slope and y intercept of 2x - 5y = 4

Solution:

Step 1: Given equation

2x - 5y = 4 ................... (1)

Step 2: Subtract 2x on both sides of the equation 2x - 5y = 4

2x - 5y - 2x  = 4 - 2x

- 5y = 4 - 2x

Step 3: Divide by (-5) on both sides of the equation

Therefore,

y = `2/5` x - `4/5`

y = 0.4x - 0.8 ............. (2)

Step 4: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 5: Compare the equation (2) with the slope intercept form of a line equation

By comparing the equation (2) with the slope intercept form, we get

m = 0.4     and b = - 0.8

Step 6: Solution

Therefore,  Slope = 0.4

y intercept = - 0.8

Example problem 3:

Determine the slope and y intercept of 9x + 3y = 0

Solution:

Step 1: Given equation

9x + 3y = 0 ............ (1)

Step 2: Subtract 9x on both sides of the equation 9x + 3y = 0

9x + 3y - 9x = - 9x

3y = - 9x

Step 3: Divide by 3 on both sides of the equation

Therefore,

y = - 3x ............... (2)

Step 2: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 3: Compare the equation (2) with the slope intercept form of a line equation

By comparing the equation (2) with the slope intercept form, we get

m = - 3     and b = 0

Step 4: Solution

Therefore,  Slope = - 3

y intercept = 0

Practice Problems to Determine the Slope and Y Intercept:

1) Determine the slope and y intercept of y = 4x - 13

2) Determine the slope and y intercept of 5x + y = 5

3) Determine the slope and y intercept of 3/2 + 2y = 4

Solutions:

1) Slope = 4; y intercept = - 13

2) Slope = - 5; y intercept = 5

3) Slope = -0.75; y intercept = 2

Non Consecutive Vertices

Non consecutive vertices mean the vertices which are not in the adjacent. Here we are going to learn about the non consecutive vertices of the planes. Generally we can say opposite vertices of the polygons are called non consecutive vertices. We will see some examples for non consecutive vertices. This will help us to understand the non consecutive vertices.  Basically a vertex in a polygon refers the intersection point of two sides.

Non Consecutive Vertices:

Basically non consecutive vertices of a polygon mean the vertex which does not lies next to the other. In other words we can say the opposite vertices or the other vertices which is not lies very next to the vertex. The distance between the two vertices will give us the height or width of the polygon. If we find the distance we can calculate the area and volume of the polygon.

If we are having a polygon with n number of vertices then each polygons vertex is having n – 3 non consecutive vertices. For example if we take a square it is having 4 sides. So for each vertex it has n – 3 = 4 – 3 = 1 non consecutive vertices. We can calculate the diagonals of the square using the non consecutive vertices.




Examples for Non Consecutive Vertices:

Example 1 for non consecutive vertices:

Find the number of non consecutive vertices in the following polygon.



Solution:

In the above diagram we are having the number of sides as 6.

We know for a single vertex we have n – 3 numbers of vertices.

So for a six vertices we have 6 (n – 3) vertices = 6 (6 – 3) = 6 x 3 =18 non consecutive vertices.


Example 2 for non consecutive vertices:

Find the number of non consecutive vertices in the following polygon.



Solution:

In the above diagram we are having the number of sides as 8.

We know for a single vertex we have n – 3 numbers of vertices.

So for a six vertices we have 8 (n – 3) vertices = 8 (8 – 3) = 8 x 5 =40 non consecutive vertices.

Negative Cosine Graph

Basically if we draw the cosine graph mean the graph will like a normal graph. Negative cosine graph is nothing but the negative sign in front of the cos.  A negative cosine equation will be like the following

Y = - a cos (bx +c)

Hare – sign is in front of the cosine graph. If we draw the negative cosine it will produce the reflection about the x – axis. So the negative cosine graph will be opposite to the positive graph.

Having problem with Cosine Law Read my upcoming post, i will try to help you.

Comparison between Positive and Negative Cosine Graphs:

Let us any cosine function y = ± Cos ((2πx + 3))

Solution:

We want to draw the graph for both positive and negative cosine function.

Here phase shift = -c / b = -3 / 2π = -`1.5 / pi`

Period = 2π / b = 2π / 2π = 1

So the positive cosine graph will be like the following



And the negative cosine graph will be like the following



To know the difference we have to combine these two graphs. So we get



From the above we know the positive cosine graph will reflect the negative cosine graph. Here the red line graph indicates positive cosine graph and the black line graph indicates the negative cosine graph.

Example for Negative Cosine Graph:

Example 1:

Draw the graph for the following function y = - 5 Cos x

Solution:

Here the amplitude is 5 and this is having a negative sign with it. So it is a negative cosine graph. Here the period is 2π and there is no  phase shift. So the graph will be



Example 2:

Draw the graph for the following function y = - 0.5 Cos (π x / 6)

Solution:

Here the amplitude is 0.5 and this is having a negative sign with it. So it is a negative cosine graph. Here the period is 2π / (π / 6) = 12 and there is no phase shift. So the graph will be

Quadrilateral Trapezoid

A quadrilateral is a two-dimensional figure created by connecting four segments endpoint to endpoint with each segment intersecting exactly two others. It also has four sides and four angles. Sum of interior angle of any polygon is (n-2) *180o. Here quadrilateral has four sides so interior angle is (4-2)*180o = 360o.Exterior angle of any polygon is 360o so the exterior angle of a quadrilateral is 360 degree.

Trapezoid


                         Trapezoid

Characteristics of Trapezoid :-

The trapezoid is a type of a quadrilateral.
A pair of opposite equal sides is known as trapezoid.
A trapezoid has unequal sides.
Normally the trapezoid has no lines of symmetry.
A trapezoid is an irregular shape.
Two trapezoids can be used to form a Parallelogram.
There will be one pair of parallel lines in a trapezoid
Perimeter of a trapezoid is to sum all the length’s of the trapezoid
Perimeter = a + b + c + B
Area of a trapezoid is ½ h ( B + b )
The sum of the adjacent angles are equal to 180o
The sum of all the interior angles are equal to 360o
Isosceles trapezoid is a type or trapezoid

Isosceles Trapezoid


                   Isosceles trapezoid

Characteristics of  Isosceles Trapezoid :-

The Isosceles trapezoid is a type of trapezoid.
If non-parallel pair of opposite sides of a trapezium is equal then it is called as isosceles trapezium.
The angles on the both sides of the base are equal
The sum of the adjacent angles are equal to 180o
There will be one pair of parallel lines in a Isosceles trapezoid
There will be one pair of opposite sides which are equal
In a Isosceles trapezoid diagonals are equal
The sum of two adjacent angles are equal to 180o
The sum of all the interior angles are equal to 360o

Calculate Angles in a Triangle

In mathematics, the triangles are nothing but the shapes which is made by 3 lines in topic geometry. In geometry, the sides are used to calculate the angles in triangles. The classification of triangles is named as Equilateral triangle, scalene triangle, Isosceles triangle, Right triangle, obtuse triangle, and acute triangle. The formula to calculate the angles in triangles are given below.

Calculate Angles in a Triangle – Formulas:

In geometry, there are different types of formulas to calculate the values of angles in the triangles. To calculate the angles value of right angle triangles we are having certain formulas based on the sides. Let as consider x, y and z are the three sides of the right angle triangles and X is the angles of the right angle triangles. Then Pythagoras theorem is used to calculate the value of angles in right triangles.

The Right Angle Triangles are shown below:



Formulas to calculate the angles of the right triangles are given as,

Sin A = x/z

cos A = y/z

Tan A = x/y

Formula to calculate the sides of the right triangles is given as,

x2 + y2 = z2

Calculate Angles in a Triangle – Examples:

Example 1: Calculate the angles A using the given data x = 5 and y = 8?

Sol:

From the given data x = 5 and y = 8

The diagram is given as,



Formula to calculate the sides of the right triangles is given as,

x2 + y2 = z2

52 + 82 = z2

25 + 64 = z2

z2 = 89

z = 9.4

Formulas to calculate the angles of the right triangles are given as,

Sin A = x/z

Sin A = 5/9.4

Sin A = .53

A = Sin-1 (.53)

A = 32.13

Example 2: Calculate the angles A using the given data x= 8 and z = 14?

Sol:

From the given data x= 8 and z = 14

The diagram is given as,



Formulas to calculate the angles of the right triangles are given as,

Sin A = x/z

Sin A = 8/14

Sin A = .57

A = Sin-1 (.57)

A = 34.84

Calculate angles in a triangle - Practice problem:

Problem 1: Calculate the angles A using the given data x= 6 and y = 17?

Answer is given below:

z = 18.02

A = 19.26

Problem 2:Calculate the angles A using the given data x= 5 and z = 14.6?

Answer is given below:

A = 11.86

Frequency Table Values that are on a Limit

A frequency table deals with the process of data that count the number of times for each set of data, the value of count with data process in a set is called as a frequency table. It has presented in a particular value of data and the table has a count of values that shows in the raw data. This is a process which can create a frequency table in a typically raw data. Let us see about the frequency table values that are on limits.

  

Frequency Table Values that are on a Limit:

The frequency table has different types which has follows.

• Frequency table of angular:

      It is define in the rate of alteration and orientation slant in the phase of a sinusoidal waveform.

• Frequency table of grouped:

      It is use to forming a group of data information in the given values.

• Frequency table of cumulative grouped:

      A cumulative frequency grouped table is a sum of frequencies in the group table satisfying the given conditions which define the entire of all grouped frequencies in a frequency distribution.

• Frequency table of simple:

       It is a function of forming a simple data collection in the given values.

• Frequency table of cumulative:

       A cumulative frequency table is the summation of frequencies that satisfying the given conditions which describe the entire of all frequencies in a frequency distribution.

Example Problem for Frequency Table of Limit Value:

Find the following table of showing the marks assignments in a set of a 10thclass of 20 students.

2	7	5	2	7	8	2	5	9	7
4	10	6	8	3	9	5	6	4	3

To set this data of information in a frequency table.

Sol:

Step 1:

        To set given marks of data in the frequency table and create a set of two rows and three columns in them with the marks listed in them.

Marks	Tally	Frequency
2           3           4           5           6           7           8           9          10

Step 2:

We can enter the highest mark of the students from the list in the table as follows.

Marks	Tally	Frequency
2            3            4            5            6            7            8            9           10	/              /              /

Step 3:

            We can find the student marks data of all information in the table. We can find the information of the frequency in the given marks of the students as shown.

Marks	Tally	Frequency
2              3              4              5              6              7              8              9             10	///             //             //             ////            //            ///            //           //            /	3             2             2             4             2             3              2              2               1

These are the brief on frequency table values that are on limits.

Percent Return Formula

In math, how much of parts done in every hundred is called as percents. The percents are represented by the symbol ‘%’. In other words, how much of value is noted out of hundred in experiments. The formula is returned with 100. Now we are going to see about percent return formula.

Explanations for Percents Return Formula in Math

Percents return formula:

                               The percents are represented as fraction with percentage symbol that is 32/100%. We can denote the percents in whole number also like 32%.T he formula for returns the percents are P = ( observed value / total value) x 100.   

How to return the percents using formula:

                             The formula for percents is divide the observed value and total value. Then multiply the 100 with that resultant value. Now, we can say this value is percents with symbol ‘%’. Sometimes, the formula returns the decimal value.   

                

How to returns the fraction into decimal value:

                               We can represent the percent value in fraction and if there is any possible, we can simplify the fraction. Then divide the numerator value with denominator value.

More about Percents Returns Formula

Example problems for percents return formula in math:

Problem 1: Return the percent value using formula for given expression.

The student got the marks 140 out of 200. What is the percent value of student?

Answer:

The percent return formula is P = (observed value / total value) x 100.

The observed value is 140.

Return the percent as (140/200) x 100 = 0.7 x 100 = 70%.

Therefore, the formula returns the percent value as 70%.

Problem 2: Return the percent value using formula for given expression.

The fruit seller has 1650 apples out of 300 fruits. What is the percent value of apple?

Answer:

The percent return formula is P = (observed value / total value) x 100.

The observed value is 165.

Return the percent as (165/300) x 100 = 0.55 x 100 = 55%.

Therefore, the formula returns the percent value as 55%.

Exercise problems for percents return formula:

1. Return the percent value using formula for 65/130.

Answer: The percent value is 50.

2. Return the percent value using formula for 87/150.

Answer: The percent value is 58.

Function Generator Application

Function is one of a concept in mathematics. Function is a relation such that every element of a known set is connected with an element of another set. A function generator is automatically generates the output value when we enter the input in a generator.  In online, few websites are providing function tutoring. In this article we shall discuss for used function generator.

Application of function generator:

Draw graph for the given functions

Evaluate the given functions

Find the output of given functions.                 

Sample Problem for Function Generator Application:

Function generator application problem 1:

Evaluate the given trigonometry functions f(x) = - sin x and make the graph for the given function.

Solution:

We are going to locate the points of the given equation and make the graph. In the first step we take f(x) as y, we get

       f(x) = -sin x

          y = -sin x

In the above equation we put x = -2

       y = -sin (-2)

       y = 0.909

In the above equation we put x = -1

       y = -sin (-1)

       y = 0.84

Like this we find out the plotting points. From the values we get the following values

X	-2	-1	0	1
y	0.909	0.84	0	-0.84

Graph:




Function Generator Application Problem 2:

Evaluate the given trigonometry functions f(x) = -cos x and make the graph for the given function.

Solution:
We are going to locate the points of the given equation and make the graph. In the first step we take f(x) as y, we get

       f(x) = -cos x

          y = -cos x

In the above equation we put x = -2

         y = -cos (-2)

         y = 0.41

In the above equation we put x = -1

         y = -cos (-1)

         y = -0.54

Like this we find out the plotting points. From the values we get the following values

X	-2	-1	0	1
y	0.41	-0.54	-1	-0.54

Graph:

Graph of Tanx

The tanx mean that the slope of function.that is tanx = sinx/cosx or rise/run.We know that tanx = sin x/cosx, tanx can be defined for all the values of x for which cosx ≠ 0.I.e. All the real numbers except odd integer multiples of π/2 (tanx is not obtained for cosx = 0 and hence not defined for x, an odd multiple of π/2).

More about Graph of Tanx:

Tangent Function Graph(y =tanx):

• In a graph of tanx 1 cycle takes place between the -90 degree to 90 degree.
• Each cycle contains are vertical asymptotes at the end.
• The period of tanx is `Pi`
• Tanx has no amplitude. That is graph that goes infinitely in vertical direction.
• The range(R) of the function tanx = ( - `oo` ,`oo`
• The domain of the function tanx is R - {(2k+1)`pi`
• The tanx is undefined at the value of x = 90° and x = 270°.
• The tanx graph crosses the x-axis three times in the interval
• 0° ≤ x ≤ 360°. At x = 0°, x = 180°, and x = 360°, tanx = 0.
• The maximum value of the function  tanx is `oo`
• The minimum value of the function  tanx is −`oo`

Consider the table for tanx between 0 to 180 degree

Graph:

The following graph shows the tanx function.

    


Problems Based on Graph of Tanx:

1.Sketch the graph for the function  y=10tant.

Graph:

Given y = 10 tant.

That is the radius of the circle is 10.The graph can be drawn by using the circle with radius 10.Let us see step by step process.





Problem 2:

Draw the graph for the function y = 25 tan t

Graph:



Practice Problem for Graph of Tanx:

1.Sketch the graph for the function y =17 tanx  by using the graphing calculator.

2.Sketch the graph for the function y =40 tanx  by using the graphing calculator.

3.Sketch the graph for the function y =33 tanx  by using the graphing calculator.