Sketching a Polynomial Graph

To some people, sketching linear or quadratic equations appear to be a piece of cake, but when it comes to sketching polynomial graphs, we’re talking about “a whole nother” level. 

Today, we’ll be running through the process of sketching a polynomial division graph.

First, go ahead and familiarize yourself with the word “polynomial”. As a mathematical term, polynomial refers to an expression of more than two algebraic terms. 

Here comes the sketching procedure:

  1. Factor the polynomials
  2. Find Vertical Asymptotes
  3. Find Horizontal Asymptotes
  4. Solve for Y-intercept
  5. Solve for zeros

We will use the expression below as an exemplar. 

2x

Why Tan θ = sin θ / cos θ

Frequently in math, when it comes to trigonometry, a lot of people know that Tan θ = sin θ / cos θ, but do you actually know why this concept is logical?

Let’s take a look at this triangle inside of the semi circle.

The triangle has a hypotenuse of the same length as that of the radius of the circle. We represent the opposite side of θ as y (the vertical axis) and the adjacent side as x (the horizontal axis). With that we can find cos θ and sin θ using x and y .

SOH CAH TOA

Sin θ = Opposite / Hypotenuse = y / 1 = y

Cos θ = Adjacent / Hypotenuse = x / 1 = x

Tan θ = Opposite / Adjacent = y / x = sin θ / cos θ

 

Flame Test Lab

FLAME TESTS LAB

Objective: To observe the relationship between various elements and their emission spectrum.

Introduction: Flame tests provide a way to qualitatively test for the presence of specific elements by seeing colored flames.  The heat of the bunsen burner excites the electrons in the atom, and this energy is released as the electrons “fall back” to their ground states. The color we see is a combination of the visible wavelengths of light emitted by the atoms.  

If you have a spectroscope you can also make quantitative observations. A spectroscope can be used to see a pattern of narrow lights called an emission (bright-line) spectrum.  The actually wavelengths of the spectrum serve as a quantitative test to determine atoms identities.  Each element has a different “pattern” of electrons so it will show a different combination of colors.  

In this lab, you will perform flame tests on seven different elements. You will use your observations to identify an unknown solution.  

Prelaboratory Questions:

1, Why do we see colors in the flame tests?

We see colors in the flame tests because of the heat from the flame excites the electrons and causes it to move to another shell, the energy is released once the electrons “fall back” to their ground state. The color we see is a combination of a visible wavelength that produces the visible light that we see. Each element has a different “pattern” of electrons. So the color combination may be different.

2, How will we be testing the substances qualitatively?

We will be testing the substances qualitatively by observing the flame colors.

Materials: Safety goggles, wood splints, tongs/tweezers, Bunsen burner, test tubes with various compounds

Safety:

1, Safety goggles must be worn at all time

2, Many of these salts are toxic. If you come into contact with any of the compounds make sure to notify the teacher and wash the contacted area thoroughly. Wash your hands before leaving the lab!

Procedure:

1, Light the Bunsen burner (turn the gas on so you can just hear it, then use the striker)

2, Place the wood splint for each compound into the flame using tongs or tweezers- ONE AT A TIME!

3, Take note of the color of the flame and return the wood splint to the solution.

4, CLEAN UP YOUR STATION! Carefully put the stoppers back on the solutions! Make sure the station looks like it did when you started! Let me know if you need new splints!

5, Wash your hands thoroughly before leaving the laboratory

Data Table: make a section of your lab labeled Data Table and make a data table similar to the one below to record your observations.

Compound The color of Flame (qualitative) Wavelengths of light (in Å) (quantitative)
Barium Chloride Orange 5900 – 6200A
Calcium Chloride Red 6200 – 7500A
Copper (II) Chloride Green 4950 – 5700A
Lithium Chloride Violet 3800 – 4500A
Potassium Chloride Orange 5900 – 6200A
Strontium Chloride Red 6200 – 7500A
     

Discussion and Analysis: (In a section labeled Discussion and Analysis answer the following questions in complete sentences)

1, How do your results from the flame test provide support for quantized energy levels? Explain your answer.

By heating the compound, the electrons got excited and jump to a higher energy. At the electrons “fall back” to their ground state, they emit energy  (photon). The different element has a different pattern of electrons, therefore it has a different energy level. While releasing the energy, they also produce visible light as a result of a combination of visible wavelengths. This allows us to see the colors of the flame (qualitative data). The result collected from the flame test provide support for quantized energy levels as it provides the color for different compound and allowed us to look up the wavelength of different light color.

 

 

 

របៀបតែងបទពាក្យប្រាំពីរ

បទពាក្យ៧

ក. រង្វាស់កាព្យ​ : 

ក្នុង១វគ្គ មាន៤ឃ្លា ក្នុង១ឃ្លាមាន ៧ព្យាង្គ្គ​ ។

ខ. ចួនក្នុងវគ្គ : 

ព្យាង្គ្គទី៧ នៃឃ្លាទី១​ ចួននឹងព្យាង្គ្គទី២ ឬព្យាង្គ្គទី៤ នៃឃ្លាទី២ ។

ព្យាង្គ្គទី៧ នៃឃ្លាទី២ ចួននឹងព្យាង្គ្គទី៧ នៃឃ្លាទី៣ ហើយចួននឹងព្យាង្គ្គទី២ ឬព្យាង្គ្គទី៤ នៃឃ្លាទី៤ ។

គ. ចួនឆ្លងវគ្គ: 

ព្យាង្គ្គទី៧ នៃឃ្លាទី៤ ក្នុងវគ្គមុន ចួននឹងព្យាង្គ្គទី៧ នៃឃ្លាទី២ ក្នុងវគ្គបន្ទាប់ ។

ឃ. កំណត់សម្គាល់: 

បើសិនជាកំណាព្យចួន​នឹងឃ្លាទី២ រាល់វគ្គក្នុងកំណាព្យត្រូវចួននឹងឃ្លាទី២ ។ ហើយបើសិនជា កំណាព្យចួននឹងឃ្លាទី៤ រាល់វគ្គក្នុងកំណាព្យត្រូវចួននឹងឃ្លាទី៤

ឧទាហរណ៍: 

(និពន្ធដោយ ឈឺន ឈឿ)

ចួនក្នុងវគ្គ

ព្យាង្គ្គទី៧ នៃឃ្លាទី១ ក្នុងវគ្គទី១  ​​ ជួននឹង ព្យាង្គគ្គទី៤ នៃឃ្លាទី២ ក្នុងវគ្គទី១ ។

ព្យាង្គ្គទី៧ នៃឃ្លាទី២ ក្នុងវគ្គទី១   ជួននឹងព្យាង្គទី៧ នៃឃ្លាទី៣ ក្នុងវគ្គទី១ ហើយជួននឹង

ព្យាង្គទី៤ នៃឃ្លាទី៤ ក្នុងវគ្គទី១ ។

ចួនឆ្លងវគ្គ

ព្យាង្គ្គទី៧ នៃឃ្លាទី៤ ក្នុងវគ្គទី១   ជួននឹងព្យាង្គទី៧ នៃឃ្លាទី២ ក្នុងវគ្គទី២ ។