I warmly welcome every steemian in week 6 course about algebra!
It's important to discuss introduction of algebra and important terminologies about algebra before getting involved into in depth of main topic that I am teaching you guys today!
Algebra is basically a major branch of mathematics which consists of study of variables and their relationships with eachother.If I talk about algebra then It includes usage of different symbols, equations, and formulas for solving problems and for modeling real world applications.
Graphing functions
If I talk about graphing functions then it's a significant concept in mathematics which permits us in visualizing and analyzing relationships existing among variables,Graphic functions may be linear, quadratic as well as exponential so let's have a look at them!
Linear functions
If I talk about linear functions then they have constant rate of alternations which result in formation of straight line graph.General form of linear functions is f(x) = mx + b in which m is slope and b is y-intercept.
• Linear functions graphs may be represented as straight lines.
Quadratic functions
If I talk about quadratic functions then they have squared variable which results in formation of parabolic graph.General form is f(x) = ax^2 + bx + c.Its graph may remain open in upward or downward and it depends at sign of a.
• Quadratic functions graphs may be represented as parabolas.
Exponential functions
If I talk about exponential functions they they contain base raised to a power which results in formation of curved graph.General form may be represented as f(x) = ab^x.Its graph may increase and decrease quickly which depends at value of b.
• Exponential functions graphs may be represented as curved lines.
If I talk about conic sections then these are those geometric shapes which originates from cone intersection and plane. There are multiple types of conic sections in which there are circles, ellipses, parabolas hyperbolas so let's discuss them briefly;
1. Circles
Circle consists of set of points which are equal from central point referred to as center.If I talk about equation for a circle then it may be written as;
(x - h)^2 + (y - k)^2 = r^2
In this equation (h, k) are circle center and r is radius.
2. Ellipses
Ellipse consists of set of points for which is sum of distances at two points which are fixed and it is constant.If I talk about equation of an ellipse then it may be written as;
(x^2/a^2) + (y^2/b^2) = 1
In this equation a and b are lengths of semi-major as well as semi-minor axes.
3. Parabolas
Parabola is set of points which are equal from fixed point which is(focus)and fixed line which is(directrix).If I talk about equation of a parabola then it may written as;
y = ax^2 + bx + c
In this equation a, b and c are constants.
4. Hyperbolas
Hyperbola is set of points in which difference of distances at two fixed points (foci)remains constant.If I talk about equation of hyperbola then it may written as;
(x^2/a^2) - (y^2/b^2) = 1
In this equation a and b is length of semi major and semi minor axes.
Identification of conic sections
For identification of conic sections there's a need of following the steps below;
Coefficients of x^2 and y^2
If both coefficients are positive then it will be ellipse or circle.Suppose if one coefficient is positive and other coefficient is negative then it will be representation of a hyperbola.If one variable is squared then it's representation will be in form of parabola.
Analyzing conic sections
For analyzing conic section there's a need of considering the following things I have mentioned below;
• There's a need of identification of center or vertex of conic section.
• Determination of orientation and length of axes.
• Location of foci of conic section.
• Identification of directrix of parabola.
• Calculation of eccentricity of ellipse or hyperbola.
For write an equation for conic section there's a need of specifying its characteristics like center, vertices, foci and axes.General forms of them are written below as;
Circles
(x - h)^2 + (y - k)^2 = r^2
In this equation (h, k)are showing center and r is radius.
Ellipses
(x^2/a^2) + (y^2/b^2) = 1
In this equation (0, 0) is representing center, a is presenting semi-major axis and b is presenting semi-minor axis.
Parabolas
y = a(x - h)^2 + k
In this equation (h, k) is presenting vertex.
Hyperbolas
(x^2/a^2) - (y^2/b^2) = 1
In this equation (0, 0)is presenting center, a is presenting distance from center to vertex and b is distance from center to co-vertex.
By specifications of these characteristics we may write equation for desirable conic section.
For solutions of problems with graphing and conic sections you need to follow the following steps;
• First there's a need of reading the problem and statement carefully.
• Secondly there's a need of graphing conic sections by use of given equation.
• Thirdly there's a need of analyzing graph for identification of unique features.
• Now there's a need for solving graph and analysis for solution of problem.
• Last but not least there's a need for checking solutions for giving surety that is it satisfying the conditions or not.
Practical example
Suppose there's a need of finding equation of circle which passes by points (1, 1), (2, 3), and (3, 2)
If I talk about solution for this question then there's a need for following these steps;
• There's a need of graphing these points.
• There's a need of analyzing of graph for finding out of center and radius.
• There's a need of using center and radius for writing equation of circle.
Equation may be written as (x - 2)^2 + (y - 2)^2 = 1
If you will follow steps I have mentioned above then easily you will solve graphing and conic sections problems!
If I talk about real world implementation and applications of graphing and conic sections then these I am describing below;
Physics and Engineering
• If I talk about conic sections then these are very helpful in modeling trajectory of projectile like path that is followed by a ball which is thrown.
• If I give another example then it may be of optical mirrors like lenses which have usage in telescope and microscope for focusing light.
Architecture and construction
• If I talk about even distribution of weight then for this purpose there are parabolic arches which are used in constructing bridges.
• If I talk about elliptical domes then they are useful in providing a most suitable structure for building.
Computer graphics and game development
• If I talk about game physics then here conic sections are utilised for stimulation of realistic motion and collision in games.
• If I talk about creation of 3D models then here graphing and conic sections are widely used.
Allah these examples are illustrating importance of graphing and conic sections in solution of real life problems.
| • | Task 1 |
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• Differentiate between quadratic and exponential functions with examples and general forms of each!
| • | Task 2 |
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• Provide real world examples of exponential functions and quadratic functions (Minimum 2 for each)that are not discussed in class!
| • | Task 3 |
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• Please find out equation of the circle with center (2, 3) and radius 4.
• Please solve for x: 2x^2 + 5x - 3 = 0
(You are required to solve these problems at paper and then share clear photographs for adding a touch of your creativity and personal effort which should be marked with your username)
| • | Task 4 |
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Scenario number 1
Suppose there's a satellite dish which is shaped like paraboloid.If equation of the paraboloid is given by y = 0.1x^2 then here x and y are measured in meters so now;
a) You need for graphing this equation.
b) You need for finding out focal length of paraboloid.
c) You need for finding out equation of directrix.
(Solve the above scenerio based questions and share step by step that how you reach to your final outcome)
Scenario number 2
You need for suppose that there's a ball which is thrown in upward direction from ground with an initial velocity of 50 ft/s.Suppose if height of ball above ground is given by the equation h(t) = -16t^2 + 50t and here t is presenting time in seconds then;
a) You need for graphing this equation.
b) You need for finding out maximum height reached by this ball.
c) You need for finding out time which it is taking for ball for reach to ground.
Follow the below mentioned rules for participating in this learning challenge!
• You can post from your own blog only.
• Your content must be original and must be steemexclusive.
• Your post should contain minimum of 500 words.
• AI generated content or plagiarized content is strictly prohibited!
• You can participate irrespective of your #clubstatus
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• Your title should be - "SLC-S22W6//Graphing and conic sections."
Tags should be - #algebra-s22w6, #learningchallange , #clubstatus , #countryname , #steemexclusive , #learnwithsteem etc
Contest deadline is from 20th Janauary to 26th Janauary.
Thank you for participating in s22 w6 learning challange!
| Criteria | Status |
|---|---|
| Plagiarism 🆓 | ✅ |
| AI 🆓 | ✅ |
| Category | Remark |
|---|---|
| Task 1 | /1.5 |
| Task 2 | /1.5 |
| Task 3 | /3 |
| Task 4 | /3 |
| Post quality | 1 |
| Total grade | /10 |
Teacher's Observation
| Task 1 review | ----- |
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| Task 2 review | ----- |
| Task 3 review | ----- |
| Task 4 review | ----- |
Recommendations or suggestions for improvement of content
Winners will be selected solely at ratings that they receive according to efforts they put in their post.All posts will be evaluated before 20 hours after posting and ratings will be clearly defined according to performance of each task by participants!At end of week,I will make sure to select top 4 users with their excellent performance in respective week.Number of comments will not be any consideration for choosing winners!
Steemcurator01 and steemcurator02 will distribute their voting power effectively throughout the week but their votes at each participant's post will not be guaranteed!
Top 4 selected users will be eligible to receive extra prize vote from steemcurator01.
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@steemchiller
Dear teacher, this week's task number 4 was very nice, with very creative questions.This week's tasks are even better for me and I will happily try to solve them, InshaAllah.
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