Quadratic Relationships

By: Karanbir Dhanki

Unit 1: Graphing Vertex Form

Learning Goals for Unit 1

Learning Goal: To learn the vertex form of graphing and all of the components.

Success Criteria: I am able to

1. Identify and define parts of a parabola

2.Graph transformations of Quadratic Relations

3.Use the quadratic function (y = x2), vertex form (y = a(x - h)2 + k),mapping notation(x+h,ay+k), and the step method (1,3,5)

4.Write equations for given parabolas

5.Identify whether a graph is quadratic, linear, or neither.

6. Find the y and x intercept

Summary of Unit

In this unit our class has learned many things regarding Quadratics and graphing using vertex form. First we learned how to tell if a graph was linear or quadratic. If a graph has the same first differences it is linear, but if the graph has the same second differences the graph is quadratic. Next, we were introduced to the parabola which are curved line with an opening either upward or downwards,the result of a quadratic function. Parabolas have many important features: vertex, axis of symmetry, y-intercept,x-intercept,minimum and maximum value. The vertex is the minimum/maximum point on the graph,where the graph changes direction(x,y). The minimum and maximum value is the highest or lowest y coordinate of the vertex (y=). The axis of symmetry is the line which divides the parabola into 2(x=).The y intercept is where the parabola hits the y axis (y=)(optimal value), and the x intercept is where the parabola hits the x axis (x=)(zero). We then learned Vertex Form: y = a(x-h) ^2 + k, which provides key information that allows us to graph the parabola. The value of a represents the shape, if -1<a<1 the parabola is vertically compressed and if a>1 or a<-1, the parabola is vertically stretched.The value of h represents the horizontal shift, h>0, the vertex moves right h units, and if h<0 the vertex moves left h units. When putting h into the vertex, always switch the sign to the opposite of what is given in the equation. This is done because in the equation h is negative. The value of k represents the vertical position of the parabola as, k>0, the vertex moves up k units and if k<0 the vertex moves down k units. To graph the vertex form we learned the mapping notation which is (x+h,ay+k) and the 1,3,5 step method. Using what we learned in this unit we were able to do many things involving finding the y and x intercepts, finding equations of parabolas, graphing parabolas and transformations and many more things. Therefore, this unit was a great base to start off quadratics, and I am excited to learn the other types of quadratic forms!

Graph

In the image below there is a graph of a equation in vertex form. To graph a parabola there are three methods. The first method is using a table of values with x and y values. You would plug in the values and then plot the points. The next method is the mapping notation method (x+h, ay+k). You would take the basic quadratic function ( y=^2) points and plug it into the mapling notation. From here you can graph the points with all of the transformations. The last method is the step method, using the pattern, going over 1 each time and up 1,3,5. You can use any of these three methods and be able to graph in vertex form.
Big image

Word problem using Vertex Form

Example: Flight Path of an Object Word Problem:

The path of the football is modelled by the relation h=-5 (t-3)^2+46.5. Where h is the height of the ball in meters, above the ground, and t is the time in seconds,since it was thrown.The questions below along with the answers, how the answers are found are shown in the pictures below.


a)Sketch the path of the football. (a: in picture below).

b) What is the maximum height of the ball? (a: 46.5m, work shown below).

c) How long does it take the ball to reach its maximum height?(a: 3 seconds, work shown below).

d) What was the initial height of the ball when it was thrown?(a: 1.5m, work shown below).

e) How long was the ball in the air?(a:6.05 seconds, work shown below).

f) What is the height of the ball at 1 second?(a:26.5m, work shown below).

Video of Graphing Using Transformations

Graphing using transformation (Vertex Form)

Unit 2: Factored form

Learning Goals for Unit 2

Learning Goal: To learn the different types of factoring and be able to graph

Success Criteria: I am able to

1.Factor different types of equations

a. Common factoring

b.Grouping

c.Simple trinomial factoring

d.Complex trinomial factoring

e.Difference of squares

f. Perfect square

2.Find out x intercepts, AOS, and vertex using factoring

3.Successfully use the 3-point method to graph a equation

Summary of Unit

During this unit we learned a variety of things revolving around factored from [ y=a(x-r)(x-s) ]. We started off with the basics as we learned about multiplying binomials and expanding and simplifying.When multiplying binomials, the two variables in first bracket are multiplied by the variables in the second bracket. After this step which is also known as expanding, we then have to simplify by collecting like terms. The next lesson we learned was common factoring and grouping. To do common factoring we have to find the GCF of the polynomials terms and factor them fully. For grouping, there are 4 terms and we have to group the first two terms and the last two terms together and find the GFC between them and then we simplify. Next we learned about simple trinomial factoring and complex factoring [(x^2+bx+c = (x+r)(x+s) ]. The difference between simple and complex factoring is in simple factoring the coefficient is always 1 (x^2). So to factor these we have to look for 2 numbers that add up to b and that multiply to equal c. To factor complex trinomials we have to multiply the coefficient of a by c. Then we have to find out 2 numbers that add up to equal b and that multiply to equal the new c. Then we are left with 4 terms which we group together and factor fully. Next, we learned about two special cases- difference of squares and perfect trinomial squares. Difference of squares is a^s - b^s = (a+b)(a-b), where you have to square root a binomial to get this equation. Perfect square trinomials are a^2 + 2ab+ b^2 = (a+b)^2 or a^2 - 2ab +b^2 = (a-b)^2. Lastly we learned the there point method by finding the zeros, AOS, and the vertex. By learning this we could graph an equation and we could also solve word problems. Therefore, this unit was very beneficial because we learned about factoring which is very helpful.

Word Problem using Factored Form

Example:Flight Path of an object Word Problem:

The height of a toy rocket launched can be approximated by the formula h= -2x^2+10x+12,where "x", is the time in seconds,and h is the height in metres. The questions below along with the answers, how the answers are found are shown in the pictures below.


a.)What is the initial height of the rocket? (a:12m, work shown below).

b.)How long does it take the rocket to reach its maximum height? (a:2.5 seconds, work shown below).

c.)What was the maximum height of the rocket?(a:32m, work shown below).

d.)What height it the rocket at after 2 seconds?(a:24m,work shown below).

e.)When does the rocket hit the ground?(a:6 seconds,work shown below).

f.)How far from the wall is the rocket when it was launched(seconds)?(a:1 second, work shown below).

Video of Factoring

Factoring by Karan Dhanki

Unit 3: Standard Form

Learning Goals for Unit 3

Learning Goal: To learn the standard form and how to graph.

Success Criteria: I am able to

1.Complete the Square & graph

2.Quadratic equations in the form y=a(x-r)(x-s)

3.Quadratic formula & graph

Summary of the Unit

In the last unit of Quadratics our class learned all about Standard Form. In this unit we first learned how to complete the square.This was changing a standard form equation to a vertex form equation by changing the first two terms of a quadratic relation into a perfect square while maintaing the balance of the original relation. By completing the square we can answer maximum or minimum questions as we get the k value. Next, as a short lesson we learned how to find the r,s,vertex, AOS and graphs in the quadratic form y=a(x-r)(x-s). Next, when factoring is not possible to find the x-intercepts we use the quadratic formula. The quadratic formula is used in algebra to solve quadratic equations. The general form is:when a,b,c are constants in the equation. This formula can be used when an equation is in standard form because variables correspond in the standard form and in the formula. We also learned about the discriminant which determines how many solutions there will be (solutions=x intercepts).The expression in the square root is called the discriminant. When the discriminant is a negative there are 0 solutions,when the discriminant is equal to 0 there is 1 solution,when the discriminant is a positive there are 2 solutions.We used the quadratic formula and completing the square to help solve word problems and to be able to graph equations. To solve the word problems and to graph and equation we needed the vertex. To find the x intercepts we use the quadratic formula and then we find the AOS by adding the two x-intercepts and dividing them by 2. Then we sub this number in to find y, and we have our vertex. The other way is by completing the square and changing the equation to the vertex form where the vertex is the h and k values. By having the 2 x intercepts and the vertex we can graph the equation or solve the word problem. Therefore, this unit was very easy to understand and it taught me a lot about the standard form.

Quadratic Formula Example

Big image

Completing the square Example

Big image

Word problem using the Quadratic Formula or Completing the Square

Example:Triangle side lengths

The length of one leg of a right triangle is 7cm more than that of the other leg.The length of the hypotenuse is 3cm more than double that of the shorter leg.


a.Draw a diagram (work shown below).

b.Create and equation(a:-2x^2+2x+40=0 ,work shown below).

c.Find the length of all three sides of the triangle(a: 5cm,12cm,and13cm, work shown below).

Reflection on Quadratics

The Quadratics unit was long and difficult, however by the end of it I learned many techniques and steps to help me solve equations and word problems. The three different forms of Quadratic are:Vertex, Factored and Standard form. In my opinion the first unit was the hardest because everything was very new for the students as we had never learned this before. In the beginning of the unit I understood the first few lessons like parts of a parabola, the vertex form and all its parts and the quadratic functions. As the unit went on we learned about graphing transformations and the mapping notation, which by the end of the unit I got a hand of and found it to be quite simple. As the end of the unit reached I had a clear understanding on how to graph a parabola, how to find the different types of transformations, how to find the x and y value and find the AOS. The next unit of Quadratics was factored form which I found quite easy by the end of the unit. In the start of the unit I found the first few lessons a easy as I understood the concepts of binomials and common factoring. As the unit went on I got a lot more confused because there were many different concepts and rules I got the different types of equations mixed up. Though as i practiced simple trinomial factoring,complex trinomial factoring, grouping, perfect square and difference of squares I got the hand of it. By the end of the unit I could easily solve different types of equations and word problems. I also learned how to graph easily using the three point method which was a simple concept. The last unit of Quadratics was Standard form, which in my opinion was the easiest unit. I understood the concepts of completing the square and the quadratic formula very easily and I was able to solve many problems. The only hard part in this unit was the word problems as we really had to think when we solved them and they got a little confusing. The last unit was a great way to end off Quadratics, and it pulled together all the pieces that we had been learning for the past few months. After completing this website, I have realized that I have become a lot better at Quadratics and I have actually grown to like it.

My Assessment: Standard Form Test