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How do you convert to vertex form


Standard Form to Vertex Form

In this mini-lesson, we will explore the process of converting standard form to vertex form and vice-versa. The standard form of a parabola is y = ax2 + bx + c and the vertex form of a parabola is y = a (x - h)2 + k. Here, the vertex form has a square in it. So to convert the standard to vertex form we need to complete the square.

Let us learn more about converting standard form to vertex form along with more examples.

1. Standard Form and Vertex Form of a Parabola
2. How to Convert Standard Form to Vertex Form?
3. How to Convert Vertex Form to Standard Form?
4. FAQs on Standard Form to Vertex Form

Standard Form and Vertex Form of a Parabola

The equation of a parabola can be represented in multiple ways like: standard form, vertex form, and intercept form. One of these forms can always be converted into the other two forms depending on the requirement. In this article, we are going to learn how to convert

  • standard form to vertex form and
  • vertex form to standard form

Let us first explore what each of these forms means.

Standard Form

The standard form of a parabola is:

  • y = ax2 + bx + c

Here, a, b, and c are real numbers (constants) where a ≠ 0. x and y are variables where (x, y) represents a point on the parabola.

Vertex Form

The vertex form of a parabola is:

  • y = a (x - h)2 + k

Here, a, h, and k are real numbers, where a ≠ 0. x and y are variables where (x, y) represents a point on the parabola.

How to Convert Standard Form to Vertex Form?

In the vertex form, y = a (x - h)2 + k, there is a "whole square". So to convert the standard form to vertex form, we just need to complete the square. But apart from this, we have a formula method also for doing this. Let us look into both methods.

By Completing the Square

Let us take an example of a parabola in standard form: y = -3x2 - 6x - 9 and convert it into the vertex form by completing the square. First, we should make sure that the coefficient of x2 is 1. If the coefficient of x2 is NOT 1, we will place the number outside as a common factor. We will get:

y = −3x2 − 6x − 9 = −3 (x2 + 2x + 3)

Now, the coefficient of x2 is 1. Here are the steps to convert the above expression into the vertex form.

Step 1: Identify the coefficient of x.

Step 2: Make it half and square the resultant number.

Step 3: Add and subtract the above number after the x term in the expression.

Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the suitable identity.

Here, we can use x2 + 2xy + y2 = (x + y)2.

In this case, x2 + 2x + 1= (x + 1)2

The above expression from Step 3 becomes:

Step 5: Simplify the last two numbers and distribute the outside number.

Here, -1 + 3 = 2. Thus, the above expression becomes:

This is of the form y = a (x - h)2 + k, which is in the vertex form. Here, the vertex is, (h, k)=(-1,-6).

By Using the Formula

In the above method, ultimately we could find the values of h and k which are helpful in converting standard form to vertex form. But the values of h and k can be easily found by using the following steps:

  • Find h using h = -b/2a.
  • Since (h, k) lies on the given parabola, k = ah2 + bh + c. Just use this to find k by substituting the value of 'h' from the above step.

Let us convert the same example y = -3x2 - 6x - 9 into standard form using this formula method. Comparing this equation with y = ax2 + bx + c, we get a = -3, b = -6, and c = -9. Then

(i) h = -b/2a = -(-6) / (2 × -3) = -1

(ii) k = -3(-1)2 - 6(-1) - 9 = -3 + 6 - 9 = -6

Substitute these two values (along with a = -3) in the vertex form y = a (x - h)2 + k, we get y = -3 (x + 1)2 - 6. Note that we have got the same answer as in the other method.

Which method is easier? Decide and go ahead.

Tips and Tricks:

If the above processes seem difficult, then use the following steps:

  • Compare the given equation with the standard form (y = ax2 + bx + c) and get the values of a,b, and c.
  • Apply the following formulas to find the values the values of h and k and substitute it in the vertex form (y = a(x - h)2 + k):
    h = -b/2a
    k = -D/4a

Here, D is the discriminant where D = b2 - 4ac.

How to Convert Vertex Form to Standard Form?

To convert vertex form into standard form, we just need to simplify a (x - h)2 + k algebraically to get into the form ax2 + bx + c. Technically, we need to follow the steps below to convert the vertex form into the standard form.

  • Expand the square, (x − h)2.
  • Distribute 'a'.
  • Combine the like terms.

Example: Let us convert the equation y = -3 (x + 1)2 - 6 from vertex to standard form using the above steps:

y = -3 (x + 1)2 - 6
y = -3 (x + 1)(x + 1) - 6
y = -3 (x2 + 2x + 1) - 6
y = -3x2 - 6x - 3 - 6
y = -3x2 - 6x - 9

Important Notes on Standard Form to Vertex Form:

  • In the vertex form, (h, k) represents the vertex of the parabola where the parabola has either maximum/minimum value.
  • If a > 0, the parabola has the minimum value at (h, k) and
    if a < 0, the parabola has the maximum value at (h, k).

☛ Related Topics:

  • Vertex Calculator
  • Quadratic Function Calculator

Standard Form to Vertex Form Examples

  1. Example 1: Find the vertex of the parabola y = 2x2 + 7x + 6 by completing the square.

    Solution:

    The given equation of parabola is y = 2x2 + 7x + 6. To find its vertex, we will convert it into vertex form.

    To complete the square, first, we will make the coefficient of x2 as 1.

    We will take the coefficient of x2 (which is 2 in this case) as the common factor.

    2x2 + 7x + 6 = 2 (x2 + 7/2 x + 3)

    The coefficient of x is 7/2, half it is 7/4, and its square is 49/16. Adding and subtracting it from the quadratic polynomial that is inside the brackets of the above step,

    2x2 + 7x + 6 = 2 (x2 + 7/2 x + 49/16 - 49/16 + 3)

    Factorizing the quadratic polynomial x2 + 7/2 x + 49/16, we get (x + 7/4)2. Then

    2x2 + 7x + 6 = 2 ((x + 7/4)2 - 49/16 + 3)

    = 2 ((x + 7/4)2 - 1/16)

    = 2 (x + 7/4)2 - 1/8

    By comparing this with a (x - h)2 + k, we will get (h, k) = (-7/4, -1/8).

    Answer: The vertex of the given parabola is (-7/4, -1/8).

  2. Example 2: Find the vertex of the same parabola as in Example 1 without converting into vertex form.

    Solution:

    The given parabola is y = 2x2 + 7x + 6. So a = 2, b = 7, and c = 6.

    The x-coordinate of the vertex is, h = -b/2a = -7/[2(2)] = -7/4.

    The y-coordinate of the vertex is, k = 2(-7/4)2 + 7(-7/4) + 6 = -1/8

    Answer: We have got the same answer as in Example 1 which is (h, k) = (-7/4, -1/8).

  3. Example 3: Find the equation of the following parabola in standard form.

    Solution:

    We can see that the parabola has the maximum value at the point (2, 2).

    So the vertex of the parabola is, (h, k) = (2, 2).

    So the vertex form of the above parabola is, y = a (x - 2)2 + 2 . .. (1).

    To find 'a' here, we have to substitute any known point of the parabola in this equation.

    The graph clearly passes through the point (1, 0) = (x, y).

    Substitute it in (1):

    0 = a (1 - 2)2 + 2
    0 = a + 2
    a = -2

    Substitute it back into (1) and expand the square to convert it into the standard form:

    y = -2 (x - 2)2 + 2
    y = -2 (x2 - 4x + 4) + 2
    y = -2x2 + 8x - 8 + 2
    y = -2x2 + 8x - 6

    Answer: Thus, the standard form of the given parabola is: y = -2x2 + 8x - 6.

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Practice Questions on Standard Form to Vertex Form

 

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FAQs on Standard Form to Vertex Form

How Do You Convert Standard Form to Vertex Form?

To convert standard form to vertex form,

  • Convert y = ax2 + bx + c into the form y = a (x - h)2 + k by completing the square.
  • Then y = a (x - h)2 + k is the vertex form.

How Do You Convert Vertex Form to Standard Form?

Converting vertex form into standard form is so easy. Just expand the square in y = a (x - h)2 + k, then expand the brackets, and finally simplify.

How to Convert Standard Form to Vertex Form Using Completing the Square?

To convert standard form to vertex form by using completing the square method,

  • Take the coefficient of x2 as the common factor if it is other than 1.
  • Make the coefficient of x half and square it.
  • Add and subtract this number to the quadratic expression of the first step.
  • Then apply algebraic identities to write it in the vertex form.

How to Find the Vertex of a Parabola in Standard Form?

Vertex can't be directly identified from standard form. Convert standard form into vertex form y = a (x - h)2 + k, then (h, k) would give the vertex of the parabola.

How to Convert Standard Form to Vertex Form Without Completing the Square?

To convert y = ax2 + bx + c into y = a (x - h)2 + k without completing the square, just find 'h' and 'k' using the following formulas

  • h = -b / 2a
  • k = - (b2 - 4ac) / 4a

What is the Use of Converting Standard Form into Vertex Form?

Vertex form is more helpful in graphing quadratic functions where we can easily identify the vertex, and by finding one/two points on either side of the vertex would give the perfect shape of a parabola.

Vertex Form Calculator

Created by Wojciech Sas, PhD

Reviewed by Steven Wooding

Last updated: Oct 16, 2022

Table of contents:
  • How to find the vertex of a parabola? Vertex equation
  • What is the vertex form of a quadratic equation?
  • How to convert from the standard form to the vertex form?
  • Vertex form to standard form converter
  • How to use the vertex form calculator?

This is the vertex form calculator (also known as vertex calculator or even find the vertex calculator). If you want to know how to find the vertex of a parabola, this is the right place to begin. Moreover, our tool teaches you what the vertex form of a quadratic equation is and how to derive the equation of the vertex form or the vertex equation itself.

And this is not the end! This calculator also helps you convert from the standard to the vertex form of a parabola, or even the other way round, in a blink of an eye!

🔎 Want to learn more about other parabola forms? Try our parabola calculator!

How to find the vertex of a parabola? Vertex equation

The vertex of a parabola is a point that represents the extremal value of a quadratic curve. The quadratic part stands because the most significant power of our variable (x) is two. The vertex can be either a minimum (for a parabola opening up) or a maximum (for a parabola opening down).

Alternatively, we can say that the vertex is the intersection of the parabola and its symmetry axis.

Typically, we denote the vertex as a point P(h,k), where h stands for the x-coordinate, and k indicates the y-coordinate.

That's enough on the definitions. But how to find the vertex of a quadratic function? It may be a surprise, but we don't need to evaluate any square root to do so!

Whenever we face a standard form of a parabola y = a·x² + b·x + c, we can use the equations of the vertex coordinates:

h = -b/(2a),

k = c - b²/(4a).

Knowing how to find these ratios, we can move one step further and ask: What is the vertex form of a parabola?

What is the vertex form of a quadratic equation?

Intuitively, the vertex form of a parabola is the one that includes the vertex’s details inside. We can write the vertex form equation as:

y = a·(x-h)² + k.

As you can see, we need to know three parameters to write a quadratic vertex form. One of them is a, the same as in the standard form. It tells us whether the parabola is opening up (a > 0) or down (a < 0). The parameter a can never equal zero for a vertex form of a parabola (or any other form, strictly speaking).

The remaining parameters, h and k, are the components of the vertex. That's where the vertex form equation gets its name.

Additionally, it's worth mentioning that it's possible to draw a quadratic function graph having only the parameter a and the vertex.

🙋 If you want to solve a quadratic equation, Omin's quadratic formula calculator will help you with the task!

How to convert from the standard form to the vertex form?

We can try to convert a quadratic equation from the standard form to the vertex form using completing the square method (you can read more about this method in our completing the square calculator):

  1. Write the parabola equation in the standard form: y = a·x² + b·x + c;

  2. Extract a from the first two terms: y = a · (x² + b/a · x) + c;

  3. Complete the square for the expressions with x. The missing fraction is (b/(2a))². Add and subtract this term in the parabola equation: y = a · [x² + b/a · x + (b/(2a))² - (b/(2a))²] + c;

  4. We can compress the three leading terms into a shortcut version of multiplication: y = a · [(x + b/(2a))² - (b/(2a))²] + c;

  5. Remove the square bracket by multiplying the terms by a: y = a·(x + b/(2a))² - b²/(4a) + c;

  6. Compare the outcome with the vertex form of a quadratic equation: y = a·(x-h)² + k;

  7. As a result of the comparison, we know how to find the vertex of a parabola: h = -b/(2a), and k = c - b²/(4a).

That is one way of how to convert to vertex form from a standard one. The second (and quicker) one is to use our vertex form calculator - the way we strongly recommend! It only requires typing the parameters a, b, and c. Then, the result appears immediately at the bottom of the calculator space.

Vertex form to standard form converter

Our find the vertex calculator can also work the other way around by finding the standard form of a parabola. In case you want to know how to do it by hand using the vertex form equation, this is the recipe:

  1. Write the parabola equation in the vertex form: y = a·(x-h)² + k;

  2. Expand the expression in the bracket: y = a·(x² - 2·h·x + h²) + k;

  3. Multiply the terms in the parenthesis by a: y = a·x² - 2·a·h·x + a·h² + k;

  4. Compare the outcome with the standard form of a parabola: y = a·x² + b·x + c;

  5. Estimate the values of parameters: b = -2·a·h, c = a·h² + k.

How to use the vertex form calculator?

There are two approaches you can take to use our vertex form calculator:

We've already described the last one in one of the previous sections. Let's see what happens for the first one:

  • Type the values of parameter a, and the coordinates of the vertex, h and k. Let them be a = 0.25, h = -17, k = -54;

  • That's all! As a result, you can see a graph of your quadratic function, together with the points indicating the vertex, y-intercept, and zeros.

Below the chart, you can find the detailed descriptions:

  • Both the vertex and standard form of the parabola: y = 0.25(x + 17)² - 54 and y = 0.25x² + 8.5x + 18.25 respectively;

  • The vertex: P = (-17, -54);

  • The y-intercept: Y = (0, 18.25);

  • The values of the zeros: X₁ = (-31.6969 , 0), X₂ = (-2.3031, 0). In case you're curious, we round the outcome to five significant figures here.

Wojciech Sas, PhD

What do you want to do?

Vertex form: y = a(x-h)² + k

Results

Vertex form equation:

y = x²

Standard form equation:

y = x²

Characteristic points:

Vertex P(0, 0)

Y-intercept Y(0, 0)

Zero X₁(0, 0)

Check out 38 similar coordinate geometry calculators 📈

Average rate of changeBilinear interpolationCatenary curve… 35 more

Eco Man without poisons and nitrates

Food as medicine

"The biggest threat to our planet is the belief that

that someone else will save it. "

Robert Swan, modern Antarctic explorer .

Ecology is a science that cares about the conservation of the natural resources of the Earth's biosphere, in order to make human life better and longer.

THE HIDDEN TRUTH ABOUT PRODUCTS, BREAD, FRUIT AND VEGETABLES

Did you know that most foods, including baked goods, contain trace amounts of agrochemicals that are harmful to humans?

Nearly all fruits and vegetables from supermarkets have been treated with about 15 chemicals. Once in your body, they will not have the best effect on your health. For example, strawberries contain an average of almost 8 pesticides per sample.

Hormonal fungicides have been found in almost 90% of non-organic citrus fruits such as tangerines, oranges and grapefruits... And the neurotoxic insecticide (permethrin) found in spinach will be a ticking time bomb even in vegetarians.

The use of garden greens, fruits and vegetables is the basis of all healthy diets. Most followers of a healthy diet do not think about the fact that these products are dangerous to health and can do more harm than good.

If you can cut down on unhealthy vegetables and fruits, what about bread? In the memory of the Russian people there are many proverbs and sayings about bread, about the attitude towards it. Bread is the basis of life and well-being, a guarantee of future success, a symbol of human joy and happiness. "There will be bread, so everything will be." But not everything is so simple with bread in our time!

Accent

According to foreign scientists , the average residual content of pesticides in non-organic bread is 61.49%, which is much higher than the total pesticide residue in all products combined, which is approximately 40%. It is alarming that the pesticide content in bread has increased from 28.24% in 2001 to 63.43% today.

“God, forgive them, for they know what they are doing. ”

Karl Kraus, Austrian writer

Agriculture - contrary to common sense?

According to Professor Pimmenthal (Purdue University), we have ignored the natural processes in the soil - microbiology and physics, especially, we have increased the amount of fertilizer and oil-based pesticides applied too much. If at 19In the 1940s, 10-50 kg of water-soluble fertilizers were enough to apply to the fields, but now this amount has increased by almost 10 times.

Pesticide use has increased by 3000% in the last 50 years!

At first, we used fallow land, crop rotation with green manure to grow foods rich in vitamins and minerals, and in the past received the same or even higher yields than we get today with extensive use of fertilizers and pesticides. Modern science better understands the biological processes that take place in the soil, and also understands how to grow foods rich in minerals. We can reverse the downward trend in minerals and vitamins in our food. Every year of using the right biological approach to growing food resonates well with the soil and increases its productivity.

Help yourself man. L. Beethoven

So the question is... How can we make sure that foods including bread, vegetables and fruits have been grown without using any toxic chemicals that could put us in danger? How to protect your loved ones? There are not many options.

For example, buy bread at an organic food store, buy fruits and vegetables at a rural market. But the most reliable is to grow it yourself "Eco-garden without poisons and nitrates"!

Obtaining a fully mineralized soil

All diseases originate in the soil - a bold statement, but true. The quality of what we eat—vitamins, minerals, and enzymes—determines our health. The body will not be strong and the immune system strong without the essential minerals that are the building blocks of our food. This building material comes from the soil and is made available to plants and animals through the activity of soil organisms.

So, in order to provide plants with plenty of nutrients, rich taste, and no pesticides, it is necessary to make sure that the soil has a balanced amount of minerals, including trace elements and a vibrant diverse community of beneficial microbes.

The most effective way to get well-mineralized soil is to focus on abundant calcium and microbes in the soil. Most soil tests show percent of the main saturation with positively charged elements - calcium - 70%, magnesium - 15% , etc. To achieve these indicators, you need to put a lot of effort and knowledge.

One of the simplest and most affordable solutions is PRK "White Pearl Drip BioCalcium + BioMagnesium":

extract of the vegetative mass of oceanic bioflora on an organo-mineral basis. A unique phytocorrector for deficiency of Ca + Mg elements, directed for emergency action. "Canned" chlorophyll — in a natural set of 72 bioelements accumulated by cells — marine photosynthesis products. A gift from nature in perfect proportion.

Principle of action: "Plant to plant"

Unlike mineral fertilizers, all elements are in an accessible natural form, bioactive, balanced (no antagonism), do not require processing by the plant. Ca + Mg freely penetrate into the metabolic system, transported by xylem and phloem currents to all parts of plants. Organic complexes of nutrients are quickly absorbed by leaf tissues, through stomata, through ion tunnels, through the cuticle.

Fast, affordable source of bionutrients. 10 times less required!..

— Manufactured to pharmaceutical guidelines, ultrafine particle size.

— No limiting factors, i.e. no sulfates/chlorides found in most calcium-containing foods.

- High solubility and mobility.

- Low salt index (5). For comparison, calcium nitrate has a salt index of 52.5.

- Compatible with many blends.

- With added seaweed extract.

- Stable in all types of soil.

In contrast to traditional forms of fertilizers, small-cluster calcium of White Pearl Drip Ca+Mg PRK quickly moves both in xylem and phloem (with an ascending and descending flow of substances), while being evenly distributed between plant organs.

Efficiency:

- prevention of plant leaf chlorosis and fruit sunburn;

- reducing the risk of bud and fruit shedding under adverse conditions;

- prevention of the development of root rot, fruit top rot, bitter pitting and other physiological diseases;

- reduction of pesticide load due to an increase in the immune status of plants and resistance to diseases and pests;

- increase in marketability of products (caliber and size of fruits), transportability and keeping quality during storage.

“Battery density?! Is this about agriculture in general?”

Phyllis Tichinin, Hawkes Bay

Battery density is a term that has become popular in the last decade. The term describes foods and meats that are high in minerals, vitamins, and secondary metabolites. Thus, cabbages that receive a full fertilizer program are tastier, weigh more than cabbages of the same variety and size, and are likely to have higher nutrient densities.

This is the most common way for growers to check if the grapes are ready for harvest. The refractometer values ​​(numbers) are the percentage of dissolved acids - sugars, vitamins and minerals in the juice. And since this value can vary from 3% to 15-20%, of course, for your money you will want to get 5 times more minerals and choose cabbage with a value of at least 15%.

)

Control Experience
Weight 365 g Weight 430 g (+65 g or 17. 8% to control)
Brix 20.4%

N:K ratio = 1:8.5 N:K ratio = 1:13.9

, mineral composition of products and the amount of secondary metabolites. The Royal Society of Chemistry of Britain (RSC) conducted a study of food. As it turned out, there was an average decrease in vitamins and minerals in our food

by 60% . We simply do not get the same concentration of nutrients that is necessary for our health with the products.

Food as medicine

Nutrient density determines the taste, medicinal qualities and shelf life of fresh foods. Plant nutrition in agriculture involves the management of minerals, microbes and humus to ensure maximum nutrient density.

If you are looking for real food grown in balanced soils and completely free of chemical pollution. If you're looking for a meal with a forgotten taste and an extended shelf life in the refrigerator that will support, not harm, your health and well-being. Read on!

We adhere to the concept of "food as medicine", and food products using the "Eco-Garden without poisons and nitrates" technology are produced with the aim of achieving this result.

Humus and Nutrient Density

So, how do you grow high mineral, low chemical foods in a biological program?

  • We know how to enlist the support of microbes in the formation of synergistic systems for the production of agricultural products.
  • We can use humic substances to reduce the use of fertilizers.
  • We use micronutrients in foliar applications to shift the plant's metabolism for better yield, flavor, keeping quality and resistance.

In short, we focus on growing humus in the soil.

Humus is a highly complex, stable by-product of the digestion of soil organic matter by microbes. It is the most complex natural substance on earth and is very difficult to test. A practical indicator of the presence of humus in the soil is the darkening of the soil to the lower layers, the roots of plants covered with thick hairs, velvety soil and rising Brix levels of plant sap.

Yes, Brix again. The more sugar and minerals in the plant sap, the more sugar it will pump into the soil to feed the microbes in the root space. Normally, a healthy plant gives soil microbes an average of 20-50% of all sugars produced. Beneficial microbes in the soil increase the amount and type of minerals, antibiotics and enzymes that enter the plant through the roots.

More minerals increase plant productivity, root growth and humus levels, since most humus is formed by microbes from dead plant roots.

The main component of humus is glomalin, which is produced by soil fungi. Soil fungi are very sensitive to pure chemical fertilizers and pesticides and are not fast growing soil bacteria. Every time we apply pesticides or unbuffered fertilizers, we are reducing the soil's ability to create humus and nutrient rich food.

Biological agriculture is a concept and technology for gradually increasing the amount of humus in the soil to produce more mineral rich food and improve environmental regeneration.

Our path to profitable farming with environmental benefits lies in the direction of fruits and vegetables rich in minerals and taste. The same processes and benefits apply to growing animal feed crops. To increase the nutritional value, it is necessary to grow humus in the soil. Over time, in such biological systems, crop yields will increase, proving the relationship between nutrient density and crop productivity.

PRK "Black Pearl Humus" - is a granular soil elixir for improving soil fertility with a complex of available elements.

Contributing to the conservation of soil moisture, Black Pearl Humus creates a nutrient medium for the development of microorganisms and bacteria. There is an accumulation of humus, soil fertility increases.

PRK "CHZH Humus" has a positive effect on the soil ecosystem, protects the fertile layer from caking, cracking. It inhibits the development of pathogens, has a beneficial effect on the development of beneficial soil bacteria and microorganisms.

The mechanism of action of PRK "Black Pearl Humus":

- activates the process of soil maturation in early spring;

- starts microbiological processes;

- improves the structure of the soil, it becomes loose and crumbly, but does not crack when dried;

- optimizes the pH level of the soil, increases the availability of soil nutrients for plants;

- increases the efficiency of fertilizers of the mineral group by 2-3 times;

- increases soil moisture retention during the dry period, increases drought and salt resistance of plants;

- contributes to the development of a powerful root system of plants, increase the number and improve the caliber of root crops, increases the yield of marketable products;

- increases the immune status of plants.

Marketable potatoes using Black Pearl Humus PRK

Interesting

A way to increase the paramagnetism and productive potential of the soil

Paramagnetism - guide to the productive potential of your soil. This phenomenon was first described by an excellent American scientist, Professor Phil Callahan. He pointed out that the productive fertility of volcanic soils is directly related to their paramagnetic quality. In fact, the higher the reading of the PCSM meter, the less problems with the soil and the higher the profit from it.

Phil Callahan and PCSM Meter

Here's how it works. Volcanic soils serve as antennas or receivers that attract and store atmospheric energy - extra long frequency radio waves. This energy originally comes from lightning bolts, but in the atmosphere, their explosive energy passes into a more subtle and stable form. Volcanic soils not only attract and store this energy, they can convert it into tiny particles of light called biophotons . The release of these tiny particles of light into the soil effectively gives light to the roots of the plant and the armies of organisms that surround them. Light energy enhances root growth and nodule formation in legumes and stimulates positive microbes.

Available paramagnetism

PRK "Black Pearl Humus", a line of micronized suspensions of PRK "White Pearl Drip BioCalcium + BioMagnesium" contains micronized volcanic rocks.

Nutrient Density and Mineral Fertilizers

Agrochemistry has come to this to some extent, since even with a weak transport system, all highly soluble substances such as calcium nitrate and complex NPKs are simply transported with water. Although water dilutes the juice, it flows very easily due to its low density. That is why foods grown on chemical additives have a rough, watery cell structure, as well as low nutritional value and poor keeping quality.

The Many Faces of Calcium

Calcium is the most important and often overlooked mineral in the vegetable garden or home garden. The first thing to understand is the relationship between soil pH and crop nutrient density. Acidic soil consumes far fewer nutrients than soil with an ideal pH of 6.4. Most minerals are most available at pH 6.4. If you neglect calcium, you are, in fact, neglecting your health.

Calcium is king!

The sooner you get an optimal level of calcium, the sooner the minerals will become more available to plants, and the microbial community will feel great ... Calcium is always the first mineral to be corrected in the soil as it has a very strong effect on other minerals. We often refer to calcium as the "carrier of all minerals" because it directly stimulates the intake of seven other minerals. It also indirectly affects the consumption of all minerals, being the gatekeeper of cell membranes through which all minerals enter the cell itself.

Note that nitrogen goes where there is calcium. It is the basis of amino acid formation, protein chemistry, and DNA copying. As soon as nitrogen appears in the field of view of any proteins, the production of enzymes and hormones begins, and various complex systems are launched, which include such trace elements as iron, zinc, copper, manganese, cobalt, molybdenum, etc.

In the soil calcium is the element that opens up the soil itself. This allows easy penetration into the soil oxygen and easy to leave the soil CO 2 for photosynthesis (gas exchange). Simply put, calcium allows the soil to breathe efficiently.

In plants, calcium is responsible for the vigor of cells and the resulting plant resistance to external influences. It also promotes cell division, plant growth and improved crop quality. With a lack of this mineral, we observe, for example, the top rot of tomatoes and capsicum.

Calcium plays an important role in determining whether your crop is harvested by devastating microbes or sap-sucking insects. A weak cell wall is the hallmark of all pests.

Calcium affects the incidence of bacterial diseases in several ways. First, calcium plays an important role in the formation of healthy, stable cell walls. Adequate calcium levels reduce the formation of enzymes produced by fungi and bacteria that dissolve the middle layer of the lamella, allowing pests to enter and infect plants. Calcium deficiency triggers the accumulation of sugars and amino acids in the apoplast, which reduces disease resistance. Fruit tissues with low calcium content are more susceptible to bacterial diseases and physiological disorders that lead to rot during storage.

To the agronomist's library

Calcium plays an important role in nitrogen fixation and amino acid chemistry, is responsible for the balance of charges in proteins, and is especially important in cell division, which occurs in fruits or seeds immediately after pollination. Without calcium, there will be no fruits or seeds.

For example, for corn, calcium targets in foliar analysis are between 0.3 and 1.0%, and they rise as the corn approaches the heading phase, and should be even higher during kernel formation. If calcium does not reach the required level, the grains at the end of the ear do not fill up. A soybean-type crop needs double or triple the amount of calcium compared to corn to fully set the pods and avoid pod cracking, which is a common soybean problem.

Do you want all the cobs to be filled so that a pod comes out of each soybean flower? This is possible only with the interaction of boron, silicon and calcium.

  • Magnesium ( Mg ) is the king of chlorophyll Since photosynthesis requires magnesium, it is the fifth element in the biochemical sequence, in first place among all trace elements. Of course, photosynthesis is not just the production of energy by chlorophyll. Energy must be converted to produce sugars from carbon dioxide and water, which requires phosphorus to convert energy. Otherwise, the chlorophyll burns out, and the leaves turn wine-red.

    Magnesium is the central part of chlorophyll , the green pigment found inside the sugar factory (chloroplast) that produces glucose, the building block of life. Magnesium deficiency leads to a decrease in photosynthesis, which will cost us a tidy sum. Magnesium also stimulates phosphorus uptake. Magnesium deficiency reduces the yield and resistance of plants to diseases.

    The calcium/magnesium ratio is one of the most important mineral interactions in the soil, allowing the soil to breathe. And also affects the optimal availability for plants of both of these minerals. Excess of any of them seriously affects the consumption of the other. In fact, all the main cations are highly interconnected and an excess of one of them will affect the intake of all the others. That is why concept cation balance is critical for soil.

    Calcium opens the soil, while excess magnesium has the opposite effect. Soils with a lot of magnesium cannot breathe normally. On such soils, you will get bogged down in dense, sticky mud. On such soils, mud clods appear during tillage, their poor gas exchange (oxygen in and CO 2 out) reduces photosynthesis and favors pathogens that do not need oxygen to live. Soils with more magnesium need more nitrogen due to its fixation, reuse and availability of nitrogen being impaired. In order to “make money on such soils”, you first need to fix calcium/magnesium ratio .

    PRK "White Pearl Drip BioCalcium + BioMagnesium" can be effectively fertigated to correct magnesium deficiency in the root zone .

    We invite partners and specialists from research institutes interested in new technologies for plant nutrition, restoration of soil fertility, increase in yield and quality of agricultural crops, to maximize the genetic potential of modern varieties, even in risky farming areas.

    For more information, contact the specialists of AgroPlus Group of Companies LLC:

    AgroPlus Group of Companies LLC

    350072, Krasnodar Territory, Krasnodar

    st. Shosseynaya (Poplar residential area ter.), No. 2/2.

    tel.: 8 (861) 252-33-32, 8 (918) 436-36-49, 8 (918) 076-21-05.

    e-mail: [email protected]

    www.agroplus-group.ru

    The Organic Farming Union is an independent social movement. Growth in the production and consumption of healthy, organic products, training, consumer education, scientific research, the introduction of eco-agrotechnologies in the agro-industrial complex.

    Health of soils, ecosystems and people.

    Enter the Union of Organic Agriculture

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    ponimayka1 - p. 36

    Kemetal in Op. Box of the Reverse operation and in the attributes of the polyNormal node, the Normal Mode parameter. If you change it from Reverse and Extract to just Reverse, the vertices will not be duplicated when the normal is reversed. However, MAYA will still draw the boundary edges, signaling a problem.

    Select another face and flip the normal with the Reverse option.

    In this case you get really bad geometry. Bad in terms of mathematics. Two faces protrude from one edge, the normals of which are directed in different directions. Important note. The most important detector of problems and bad geometry is the conversion of a polygonal surface into subdivisions. For a quick surface quality check, use the standard Alt-` hotkey in combination with Undo. In case of an error, read the Script Editor, there will be a diagnosis and even a prescription for treatment.

    If you try to convert a plane with a bad edge to a subdiv (Modify=>Convert =>Polygons to Subdiv), MAYA will "swear", indicating that some edges are nonmanifold (nonmanifold), that is, they do not connect faces smoothly:

    //Error: line 2: polyToSubdivl (Poly To Subdiv Node): One or more edges is nonmanifold.

    In this case, the cure is obvious: you just need to reverse the normal with the same Reverse option. However, to find all such non-manifold edges on a surface, it is easier to use the universal cleaner, the Cleanup operation.

    Select the surface itself and open the Option Box of the Cleanup operation.

    Check the Nonmanifold Geometry checkbox. By default, the Normals and

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    Geometry treatment method is selected, using which MAYA first tries to unfold the normals to get rid of bad edges, and if this does not help, then duplicates these edges, making them boundary and cutting adjacent faces. If the Geometry Only option is selected, it does not touch the normals, but immediately duplicates the bad edges.

    Experiment with both options by clicking Apply and Undo. See how MAYA then converts the surface into a subdiv.

    The concept of "non-manifold geometry" can be translated as follows: a mesh that cannot be unfolded into one flat non-overlapping patch.

    I will give additional examples of non-manifold geometry.

    If you killed the faces in a checkerboard pattern, the remaining faces are joined by vertices, not edges. Such geometry also cannot be converted to a subdiv, and any anti-aliasing does not work on it, although it will be rendered and animated without problems. In this case, the Cleanup operation simply duplicates the corner vertices.

    Another example of a non-manifold mesh is the result of an Extrude Edge operation on internal edges when more than two faces protrude from the same edge. Such a design cannot

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    be unfolded into a flat flap and converted into subdivisions.

    At the same time, it can be smoothed out and shouldn't cause problems when rendering.

    The Cleanup operation in this case duplicates the problematic edge so that each face sticking out of it has its own individual edge.

    The main sources of bad geometry are unwashed user hands, chronic sleep deprivation, Extrude Edge, Normals=>Reverse, Delete Face, Collapse, Reduce operations.

    Let me give you another example, often encountered for the first two reasons.

    Create a plane.

    Copy it, as if by accident, without noticing the copy. Select both planes and do Combine.

    Like good boys, do Merge Vertices after this.

    You now have a visually decent surface that has double edges. And

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    is quite difficult to detect. By dragging the vertices, you are synchronously changing these double faces, which have all edges in common. Such faces are called lamina faces. By selecting any face and dragging it, you will simultaneously drag all the edges, and, as a result, the second face. And just by selecting a face not by clicking, but by circling its center with a box, you can notice (which is unlikely) in the title bar of the MAYA window that two faces are selected.

    Tip. Turn on the display of polygonal statistics on the screen; Disptay=>Heads Up Display. This will allow you to control how many and which components are selected at any given time.

    The treatment of such a "laminar" virus is quite intricate.

    Try to convert the resulting plane into a subdiv. You will get the standard error message (stating that the geometry is non-manifold):

    //Error: line 2: polyToSubdivl (Poly To Subdiv Node): One or more edges is nonmanifold.

    //Nonmanifold geometry cannot be converted to a subdivision surface.

    //To clean up nonmanifold edges, use Polygons->Cleanup with the nonmanifold option. / /

    Here, of course, you will call "antivirus", that is, in the Option Box of the Cleanup operation, check the Nonmanifold Geometry checkbox.

    However, this is not the option to use in this case. However, you may not know this, and therefore, by selecting the default Normal and Geometry option and pressing the Apply button, you will get a rather strange set of selected faces. If you're lucky enough to be able to show normals, you'll see that the surface has normals on both sides!

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    If you don't guess, you'll see: MAYA simply cut the surface into separate faces. If you are not completely discouraged by the result, you will probably move vertices or edges to find out what is wrong, and finally find that the surface is double!

    Once you see the double layered surface, press Undo as many times as needed to return to the "uncooked" surface.

    Open again the Option Box of the Cleanup operation, uncheck the Nonmanifold Geometry checkbox and check the Lamina Faces checkbox at the very bottom.

    The treatment will now be more effective.

    The Remove Geometry section also contains options that allow you to remove faces that are less than a certain area threshold, or edges that are less than a given value. Since this process will be completely automatic, I would advise you to use it with great care, as such a removal may cause another cleaning of the surface.

    The Tessetate Geometry section contains parameters that are not related to cleaning itself

    and surface treatment. They allow you to break the mesh into triangles, and specify the criteria for determining which faces to break: quadrilaterals, polygons, concave faces, faces with holes, or non-planar faces. This can be useful when importing into a game engine.

    In principle, if you are not going to convert the model into subdivisions, the question of what is considered bad geometry becomes quite subjective. Especially if you do not intend to intensively bend the geometry in the future.

    Of course, double vertices or faces are in any case a defect. But other types of non-manifold geometry can only cause partial problems when smoothing the surface. Therefore, if these problems do not concern you, you can not waste time solving them.

    However, the popularity of Subdivision Surfaces at the moment is such that it would be rather frivolous to dismiss them (I'm speaking mostly in the organic modeling camp). Draw your own conclusions.

    Note. Faces with holes are not "officially" considered a defect, and some operations support them. However, smoothing operations do not work correctly on them, and when converted to subdivs, holes will simply be ignored. But the most important thing is that the Split Polygon Tool does not work on such faces. Therefore, edges with holes are prohibited from use in most Polizen schools.

    Very good geometry. About the benefits of rectangles and the dangers of triangles. Frank* Polizen Masters

    I'll warn you right away: anyone involved in architecture, design or toys can skip this section. It will focus on the quality of models intended for character animation. I do not pretend to be objective, therefore, perhaps something may seem unobvious or controversial to you.

    After speaking with the Polizen masters, I discovered that absolutely all of them mystically avoid triangles. Not satisfied with the answer that triangles have a bad effect on karma, and the number "four" is a symbol of stability, I did some research of my own, which I summarized in the following material.

    There are three technical reasons why craftsmen avoid triangles and try to work only with quadrilaterals. And also - a few esoteric reasons, about which a little later. The first reason is based on the popularity of subdivs for organic modeling. It's the subdivs that really dislike triangular faces. Do the following exercise. Take the polycube. Select one of the vertices and cut it with the Chamfer Vertex operation.

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    Delete history. Duplicate the cube and move the copy to the right.

    Select the cut face and cut it into three quadrilaterals using the Subdivide operation. Then take the Split Polygon Tool and cut the three faces adjacent to the cut one in half, diagonally from the middle of the cut edge. Delete history.

    Now select both cubes and convert them to subdivs by pressing 'Alt-'. Play with the smoothness of the display by pressing "0,1,2,3".

    Press "0" and note that the cube, which had more faces, has turned into a "lighter" subdiv. This happened precisely because it was correctly divided into quadrangles.

    In addition, grids containing triangles, pentagons and other polygons are difficult to parameterize, and even generate “special” points (“asterisks”) on subdivisions, where three, five or more “isoparms” converge. These points are a potential source of problems, since all sorts of "buckling" can appear in their vicinity when deformations occur.

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    Quadrilaterals tend to be well parametrized from the very beginning because they are pieces of a rectangular grid, so subdivisions “like” quadrilaterals very much.

    To finally verify this, do the final experiment. Select both subdivs and convert them to NURBS surfaces: Modify"Convert=>Subdiv

    to NURBS.

    Now count the number of spline patches on each of the former cubes. There will be 27 of them on the first cube, and only 9 on the sawn one.

    The second technical reason is that Polyzen masters often use Renderman to render characters. And he, oh, how he does not like triangles! And not only Renderman: after all, until recently, Mental Ray, built into MAYA, could not render Subdivision Surfaces, which did not consist of only quadrilaterals, and any triangular face in the base mesh caused him allergies. Of course, now there are effective methods for rendering triangles in subdivs (such as Loop Scheme), but the sediment, as you understand, remains, and triangles still have a reputation as a hindrance to rendering subdivs.

    The third reason is not related to subdivs, but to character animation. Very often, non-quad faces located at surface folds cause unpredictable deformations. Also, in the same places, it is undesirable to have a vanishing point of several "patches", that is, a vertex from which more than four edges emerge. All this leads to unpleasant consequences in the form of "disturbances", screeds, creases, etc. Of course, if you cannot do without triangles at all, you can leave them in some places, using the following rule: the main thing is that a given non-quadrangular (triangle, pentagon, etc.) .d.) did not cause perturbations on the surface after smoothing or when translated into subdiv. And behaved decently during the animation. Practice shows that such non-quadrangular polygons cannot be left in those places that are actively stretched, compressed during animation. Especially if the character's skin is not covered with fur. Roughly reformulate this rule as follows: if you want to effectively bend or somehow deform the surface, keep its edges quadrangular. Let it need an extra chain of edges.

    There are also some existential reasons for avoiding non-quadrangles. If you look around, you can find a lot of examples that in nature, quadrilateral patches are often preferred, when talking about surfaces, not about volumes. Take, for example, fabrics: their fibrous structure is a rectangular grid. In confirmation of this, no doubt, controversial hypothesis, I would like to cite a story that one of the Polidzen masters Sergey Lutsenko told me. The story is very instructive

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    and at the same time incomprehensible to those who have not yet felt what “the form does not live” means. Read and try to feel it.

    “I made my personal discovery about how the edges and faces of the model should be distributed when I once had to model the cap of an American policeman. I got stuck on modeling the tulle (the very top of this headdress |. An attempt to just formally lift up the front of the tulle did not lead to anything good. The cap was not a cap anyway. The wife who was modeling clothes came to the rescue. She, looking from the side ", immediately pointed out the mistake and suggested paying attention exactly to how the fabric is located on a real object, what and how many pieces of fabric the cap consists of? How and where it is sewn. And when I reproduced all these fragments, exactly according to the original, the cap "came to life ". The design is working!"

    Some orthodox Polyzen masters of the old Indian school still make the initial blank of the model with spline patches, which are then converted into polygons, getting flawless edges and perfect texture mapping to boot.

    At this point I would like to finish the ode to quadrilaterals and briefly highlight three operations for splitting faces into the required number of vertices.

    Triangulate, Quadrangulate and Subdivide

    To spoil the model and turn all its faces (or not all, but selected ones) into triangles, there is a special operation: Triangulate. (This is a joke, and often this operation is needed to export the model to the game engine or feed it to the input of a program that, apart from triangles, does not understand anything - after all, there are others).

    When I started learning MAYA, I expected that the Quadrangulate operation does the same, not the opposite, that is, it finds all non-quad faces and cuts them into four vertices. But no, this operation is the opposite of Triangulate: it only searches for triangular faces, and if the angle between adjacent triangles is less than the specified one, these two triangles are turned into a quadrilateral by deleting a common edge. The process, as you know, is not very predictable.

    But if you want to ensure that the selected faces turn into quadrilaterals, you should use the Subdivide operation. She breaks the edges in the right way. In addition, it also works for selected edges.

    You can select only triangular or only quadrilateral faces using the Edit Polygons=>Selection=>Selection Constraints window. Also keep in mind that its content depends on the active components.

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    Also, don't forget: one of the added bonuses of the Smooth operation is that it only creates quad faces during the smoothing process (for the Exponent method).

    A little more about normals

    Let's talk a little about vertex normals. Let me remind you that these are normals sticking out of vertices, the number of which in each vertex is equal to the number of faces adjoining it. These normals have a decorative function and affect how soft or hard transitions will be in the edges when lit.

    Basically, if you are going to convert the surface to subdivs, vertex normals shouldn't bother you at all: in that case they are ignored. However, they often disfigure the appearance of the surface on the screen so much that it is useful to find some kind of control on them. Very often the model "arrives" in the scene with bad vertex normals when imported from other packages. In this case, it is useful to do the following procedure to treat corrupted vertex normals.

    The easiest way is to select the entire surface and perform the Edit Polygons=>Normals=>Set Vertex Normal operation, and in the Option Box, check the Unlock Normals checkbox.

    This checkbox with an illogical and incomprehensible name says that you need to not only unlock vertex normals, but also recalculate them to default values ​​(these values ​​are determined by the softness / hardness of the edges set earlier).

    After that you can set the hardness/softness of the edges with the Soften/Hard en operation. The mechanism of blocking vertex normals is not obvious, since it is NOT explicitly displayed anywhere on the screen whether vertex normals are blocked or not.

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