The post STANDARD TO VERTEX FORM CALCULATOR appeared first on Mikes Calculators with Steps.

]]> 1 – Step by Step Solver: Find the Vertex of any Quadratic Equation

2 – How do you convert from Standard Form to Vertex Form?

3 – How do you locate the Vertex on the Graph of a Parabola?

4 – Example: How do you convert from Standard Form to Vertex Form?

5 – How do I find h and k in Vertex Form?

6 – What are h and k in Vertex Form?

Next, press the button to find the Vertex and Vertex Form with Steps.

\boxed{ y=ax^2+bx+c }

Then, the Vertex (h,k) can be found from the above Standard Form using

\boxed{ h= {-b \over 2a} , k=f( {-b \over 2a }) }

Once computed, the vertex coordinates are plugged into the Vertex Form of a Parabola, see below.

The Vertex is just that particular point on the Graph of a Parabola.

See the illustration of the two possible vertex locations below:

We are given the Standard Form

y=3x^2- 6x-2 .

First, compute the x-coordinate of the vertex

h={ - b \over 2a}= { -(-6)\over (2*3)} = 1 .

Next, compute the y-coordinate of the vertex by plugging h=1 into the given equation:

k= 3*(1)^2-6(1)-2=-5 .

Therefore, the vertex is

(h,k)=(1,-5) .

Thus, we transformed the above Standard Form into the Vertex Form

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

Easy, wasn’t it?

Tip: When using the above Standard Form to Vertex Form Calculator to solve

3x^2-6x-2=0 we must enter the 3 coefficients a,b,c as

a=3, b=-6 and c=-2.

Then, the calculator will find the Vertex (h,k)=(1,-5) step by step.

Finally, the Vertex Form of the above Quadratic Equation is

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

Get it now? Try the above Standard Form to Vertex Standard Calculator again.

There are two ways to find h and k, the vertex x- and y- coordinates.

1) The **fast** way: Given y=ax^2+bx+c we first compute h= {-b \over 2a} and next k=f(h) .**Example:** y=3x^2+6x+4 thus h= {-6 \over 2*3} = -1 and k=f(-1)=3(-1)^2+6(-1)+4=3-6+4=1

Thus, Vertex Coordinates are (k,h)=(-1,1)

2) The **long** way: We do the Complete-the-Square procedure to convert

y=ax^2+bx+c into

y=a(x-h)^2+k .

Please click **HERE **to do this procedure.

h and k are the Vertex x- and y- coordinates of the Graph of a Quadratic Equation. They give the Location of a Minimum (when a>0) or Maximum (when a<0).

You may also think of h and k as shifts/transformations:

Shifting the Standard Parabola

y=x^2

h units right yields

y=(x-h)^2 .

Shifting it k units up yields

y=(x-h)^2+k .

By performing those 2 shifts we moved the Vertex from (0,0) to the new location (h,k) .

The post STANDARD TO VERTEX FORM CALCULATOR appeared first on Mikes Calculators with Steps.

]]>The post VERTEX FORMULA OF A PARABOLA appeared first on Mikes Calculators with Steps.

]]> 1 – Step by Step Solver: Find the Vertex of any Quadratic Equation

2 – How do you find the Vertex of a Parabola?

3 – What is the Vertex of a Parabola

4 – Video: How to Find the Vertex Coordinates?

5 – Solved Problem: How do you find the Vertex of a Vertex?

Next, press the button to find the Vertex and Vertex Form with Steps.

The Quadratic Formula Equation

\boxed{ ax^2+bx+c = 0 }

has the Vertex Coordinates (h,k) where

\boxed{ h= {-b \over 2a} , k=f( {-b \over 2a }) }

Once computed, (h,k) are plugged into the Vertex Form of a Parabola below.

The Vertex is just that particular point on the Graph of a Parabola.

See the illustration of the two vertex locations below.

The Vertical Line passing through the Vertex is the “Axis of Symmetry”: Reflect the one side of the parabola over this axis to obtain the other side the parabola.

We are given the quadratic equation 3x^2- 6x-2 .

First, compute the x-coordinate of the vertex

h={ - b \over 2a}= { -(-6)\over (2*3)} = 1 .

Next, compute the y-coordinate of the vertex by plugging -1 into the given equation:

k= 3*(1)^2-6(1)-2=-5 .

Therefore, the vertex of the vertex is

(h,k)=(1,-5) .

Thus, the vertex form of the Parabola is y=(x-1)^2-5 .

Easy, wasn’t it?

Tip: When using the above Quadratic Vertex Formula to solve

3x^2-6x-2=0 we must enter the 3 coefficients as

a=3, b=-6 and c=-2 .

Then, the calculator will find the vertex (h,k)=(1,-5) step by step.

Finally, the Quadratic Vertex Formula is y=(x-1)^2-5 .

Get it now? Try the above Quadratic Vertex Formula Calculator a few more times.

The post VERTEX FORMULA OF A PARABOLA appeared first on Mikes Calculators with Steps.

]]>The post VERTEX TO STANDARD FORM CALCULATOR appeared first on Mikes Calculators with Steps.

]]>Next, press the button to convert the Vertex to Standard Form of a Parabola with Steps.

y=a(x-h)^2+k

where (h,k) are the Vertex Coordinates.

The Standard form of a Parabola is

y=ax^2+bx+c

To obtain the Standard Form from the Vertex Form we use these steps:

y=a(x-h)^2+k

y=a(x^2-2hx+h^2)+k

y=ax^2-2ahx+ah^2+k

Example: To convert

y=2(x-1)^2-5

we first apply the binomial formula to get

y=2(x^2-2x+1)-5

Next, we distribute the 2 to get

y= 2x^2-4x+2-5

With 2-5=-3 we finally arrive at the Standard Form:

y=2x^2-4x-3

..

..

The Vertex is just that particular point on the Graph of a Parabola.

See the illustration of the two possible vertex locations below:

We are given the quadratic equation in vertex format

y=2(x+3)^2-7

First, apply the binomial formula

(x+3)^2 = x^2+6x+9

Thus we have

y= 2(x^2+6x+9)-7

Next, distribute the 2 to get

y= 2x^2+12x+18-7

Since 18-7=11 we finally get the standard form

y= 2x^2+12x+11

Here, a=2, b=12 and c=11 are the coefficients in the Standard Form

y= ax^2+bx+c

Get it now? Try the above Vertex to Standard Form Calculator a few more times.

The post VERTEX TO STANDARD FORM CALCULATOR appeared first on Mikes Calculators with Steps.

]]>The post Parabola Form appeared first on Mikes Calculators with Steps.

]]>Parabolas are the Graphs of Quadratic Equations.

There are 3 different forms of Quadratic Equations:

Standard Form: \boxed{ y = ax^2+bx+c }

Vertex Form: \boxed{ y = a(x-h)^2+k} . (h,k) = Vertex Coordinates.

Factored Form: \boxed{ y = a(x-r)(x-s)} . r and s = Zeros of the Parabola.

Standard Form: \boxed{ y = x^2+6x+8 } can be rewritten in as

Vertex Form: \boxed{ y = (x+3)^2-1} . It tells us that above Parabola has Vertex Coordinates = (3,-1) . The Process to convert from Standard Form to Vertex Form is called “Completing the Square” and can be done here with steps:

Solve A Quadratic Equation by Completing the Square

Factored Form: \boxed{ y = (x+4)(x+2)} . It tells us that above Parabola has Zeros -4 and -2 . The Process to convert from Standard Form to Factored Form is called “Factoring a Quadratic Equation” and can be done here with steps:

Factoring Quadratic Equations – Calculator

Or simply use the head menu when converting between the different Parabola Forms.

The post Parabola Form appeared first on Mikes Calculators with Steps.

]]>The post Domain and Range of a Parabola appeared first on Mikes Calculators with Steps.

]]> 1 – Step by Step Solver: Find the Range of a Parabola

2 – What is the Domain of a Parabola?

3 – How do I write the Range in Interval Notation?

4 – Example: Domain and Range of a Parabola

5 – Is there a Domain and Range Parabola Calculator?

Next, press the button to find the Domain and Range of a Parabola with Steps.

The reason for that is quadratic equations fall in the category of polynomials and thus don’t contain fractions, roots or radicals nor logarithms. Those 3 categories of functions have restrictions with regard to their domain.

Notice that the 3 is included in the Range so we use bracket [ in the Interval Notation. Since there is no upper bound for the Range we denote that with the Infinity Symbol which is always followed by the Open Interval symbol “)”. Think of Infinity as NOT being a concrete endpoint.

Again, the 5 is included in the Range so we use the bracket ] in the Interval Notation.

We are given the Standard Form

y=3x^2- 6x-2 .

Since ANY x value can be plugged into this equation (we have no fractions, radicals nor logarithms) we can conclude that the Domain is ‘all real numbers’. In Interval Notation: (-\infty , \infty)

To find the Range we first compute the x-coordinate of the vertex

h={ - b \over 2a}= { -(-6) \over (2*3)} = 1 .

Next, compute the y-coordinate of the vertex by plugging h=1 into the given equation:

k= 3*(1)^2-6(1)-2=-5 .

Therefore, the vertex is

(h,k)=(1,-5) .

The Range is simply all real numbers greater or equal to -5 or simply y>=-5.

Using Interval Notation: [-5 , \infty )

Explanation: This parabola opens to the top since the leading coefficient 3 is greater than 0. This implies that the vertex is a minimum and therefore the parabola takes on any value greater than 5. Since the Range contains all possible y- values the Range is [-5 , \infty)

Easy, wasn’t it?

Tip: When using the above Range of a Parabola Calculator for 3x^2-6x-2

we must enter the 3 coefficients a,b,c as a=3, b=-6 and c=-2.

Get it now? Try the above Range of a Parabola Calculator again if needed.

The post Domain and Range of a Parabola appeared first on Mikes Calculators with Steps.

]]>The post Domain and Range Calculator appeared first on Mikes Calculators with Steps.

]]>Just enter your Function and press the “Calculate Domain and Range” button. The Domain and Range will be displayed in a new window.

1) The Domain is defined as the set of all possible x-values that can be plugged into a function.

2) The Range of a function is defined as the set of all resulting y values.

1) The Domain is defined as the set of x-values that can be plugged into a function. Here, we can only plug in x-values greater or equal to 3 into the square root function avoiding the content of a square root to be negative.

Thus, domain is x>=3 .

Using Interval Notation we write: [3,\infty )

2) The Range of a function is defined as the set of all resulting y values. Here, the lowest y coordinate is y=0 achieved when x=3 is plugged in. The larger the x value plugged in the larger the y coordinate we obtain.

Thus, the range is y>=0 .

Using Interval Notation we write: [0,\infty )

1) The Domain are the x-values going left (from the smallest x-value) to right (to the largest x-value).

2) The Range are the y-values going from lowest (from the smallest y-value) to highest (to the largest y-value).

1) The Domain is all real numbers. Any number can be plugged into y=6.

2) The Range is just y=6. The lowest and highest y are both 6.

Domain and Range Calculator

The post Domain and Range Calculator appeared first on Mikes Calculators with Steps.

]]>The post What is the Mean in Math? appeared first on Mikes Calculators with Steps.

]]>
Enter your Numbers (separated by commas) & Press the Button.

The **Mean** in Mathematics is just another word for **Average**.

Mathematician denote the Mean using the symbol \overline{x} .

The Mean is found by adding up the given Numbers divided by the Number of Numbers given.

As a Formula:

\overline{x} = Sum of given Numbers / Number of Numbers = \frac{x1+x2+x3+..+xn}{n}

**Example:** We are given 2,4,9. Then, \overline{x} = \frac{2+4+9}{3}= \frac{15}{3} = 5

The Mean is one of 3 common ways to describe the center of a set of numbers besides the Median and the Mode.

For Instance, if 3 boys weigh 100, 110 and 150 pounds, then on Average (aka their Mean Weight) they weigh 360/3=120 Pounds. The 120 pounds can be seen as a representation of the weight of the 3 boys.

Notice that the Mean uses each boy’s weight. So if only boy is extra light or extra heavy it will have a significant impact on the Mean. On the contrary, the Median (described below) is designed to not get affected by unusual weights.

The Median is the Center in a sorted list of Numbers. In case of an odd number of numbers there will be a single center number. In case of an even number of numbers in the list the Median equals the average of the 2 center numbers.

**1. Example:** We are given 4,9,2. First we have to sort the list to get 2,4,9 . The Median is 2 because it is in the Center of this sorted list. We have an odd (3) numbers in our list.

**2. Example:** We are given 6,4,9,2. First we have to sort the list to get 2,4,6,9 . This list has an even number of numbers. Thus, the Median is 5 as the average of 4 and 6.

For Instance, if 3 boys weigh 100, 110 and 150 pounds, then their Median Weight is 110 Pounds.

The 110 pounds can be seen as a representation of the weight of the 3 boys.

Notice that the Median does not get affected by unusual weights.

The post What is the Mean in Math? appeared first on Mikes Calculators with Steps.

]]>The post Sample Standard Deviation appeared first on Mikes Calculators with Steps.

]]>**Standard Deviation of Sample ** s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^n (x - \overline{x})^2}

**Sample Standard Deviation** gives the average distance of your numbers to the mean of those numbers.

Example 1: A Standard Deviation of 0 means that the given set of numbers are the same since they don’t differ from their mean.

Example 2: A Standard Deviation of 1 means that the given set of numbers differ – on average – by 1 from their mean.

Bowling Example: A consistent bowler that bowls 110 and 90 games has a **lower **Standard Deviation than a bowler who bowls 50 and 150. While they each have a mean of 100 the 2. bowler scores varied much more from 100 than the first bowler.

Their Difference lies in the Denominators of their Formulas (for technical reasons):

When computing the Sample Standard Deviation we divide by n-1.

When computing the Population Standard Deviation we divide by n instead.

This is the formula for the Population Standard Deviation:

\sigma = \sqrt{ \frac{1}{n} \sum_{i=1}^n (x - \overline{x})^2}The 1 Standard Deviation rule refers to the **Empirical Rule** of Normal Distributions (aka Bell Curves). See image.

About 68% of the Data fall within 1 Standard Deviation of the Mean.

About 95% of the Data fall within 2 Standard Deviations of the Mean.

About 99.7% of the Data fall within 3 Standard Deviations of the Mean.

**Example:** About 95% of the US Population have an IQ between 80 and 120.

Reason: Mean IQ=100 and Standard Deviation=10. Thus, 95% Americans fall within 2 standard deviations, 2*10 = 20 of 100.

Note: This means that about 2.5% have an IQ higher than 120.

The post Sample Standard Deviation appeared first on Mikes Calculators with Steps.

]]>The post Standard Deviation to Variance appeared first on Mikes Calculators with Steps.

]]>

Next, press Solve to find the Variance.

Both Variance and Standard Deviation are Measures of Spread in Statistics.

The Variance is the Sum of the Squared Differences between the given Data and their Mean.

Taking the Square-Root of the Variance then gives the Standard Deviation. The Standard Deviation tells us by how much the data differ from the mean, the average distance from the mean. As formulas:

Say we are given 2,3,4 . Their mean is 3 since {2+3+4 \over 3} = 3 .

The Squared Differences between the data and mean are

(2-3)^2 + (3-3)^2 + (4-3)^2 = 1+0+1=2 .

Dividing that by the number of data points, here 3, yields s^2=2/3=0.666 as the variance.

Finally, take the Square Root of the Variance to find the Standard Deviation s= \surd (0.666)=0.81649658092 .

To find Variance we have to Square the Standard Deviation.

Reason: Standard Deviation is s , the Variance is s^2 .

To find Standard Deviation we have to take the Square Root of the Variance.

Reason: The Variance is s^2 , the Square of the Standard Deviation is s .

When computing Population Variance we divide by the Population Size N (as shown in the image above).

When computing Sample Variance we divide by the Sample Size N-1 instead (for technical reasons) .

The post Standard Deviation to Variance appeared first on Mikes Calculators with Steps.

]]>The post Mean Median Mode Range Calculator appeared first on Mikes Calculators with Steps.

]]>
Enter your Numbers (separated by commas) & Press the Button.

**Mean** = Average = \overline{x} = Sum of all Numbers / Number of Numbers in List.

**Median **= Center Number of Ordered List

**Mode **= The most frequent Number in List

**Range**=Largest – Smallest Number in List

** Variance of Population ** \sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2 , \quad \mu = Population Mean

**Standard Deviation of Population** \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2} , \enspace \mu = Population Mean

** Variance of Sample ** s^2= \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2 , \quad \overline{x} = Sample Mean

**Standard Deviation of Sample ** s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2} , \quad \overline{x} = Sample Mean

Let’s do an **Example with 7 numbers** (see right image)

a) To find the **Mean **we add up the 7 integers to get 80 and divide by 7 to get a Mean of 80/7 = 11.4 . Note: **Mean **is also called **Average**.

b) To find the **Median **we simply identify the center number to get a Median = 6. In case of 2 center numbers we will average them.

c) To find the **Mode **we simply identify the most common number to get a Mode = 1. Note: We may have 2 or more modes.

d) To find the **Range **we simply subtract the Minimum from the Maximum to get a **Range **= 42 – 1 =41.

e) **Outliers **are numbers that are way off.

The post Mean Median Mode Range Calculator appeared first on Mikes Calculators with Steps.

]]>