# 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

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

*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

## 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

## 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

## Multiplying Binomials/Expanding and Simplify In this equation we multiply the terms from the first bracket by the second bracket. This expands both of the brackets.Then we simplify by collecting like terms and ending up with the answer. | ## Common factoring The greatest common factor from this equation would be 3x. So you take 3x out of the bracket and divide each term by it. The final answer would be 3x(x+2). | ## Grouping First we group the terms that have the same factor together. Then we find the GCF and and divide each term in the terms. Finally we simplify by making two brackets. |

## Multiplying Binomials/Expanding and Simplify

## Common factoring

## Simple Trinomial Factoring In this equation -4-4=-8 (b) and -4 x -4=16(c). | ## Complex Trinomial Factoring In this equation the numbers that multiply and add to equal -12 and 20 are -10 and -2. By grouping and simplifying you get the answer. | ## Difference of Squares To answer this question you have to square root both of the terms. We know this is a perfect square as there is a negative sign between the numbers, not adding .The answer is going to be one equation subtracting and one adding as it is difference of squares. |

## Complex Trinomial Factoring

## Perfect Square Trinomial This picture shows us how to factor a perfect square trinomial. | ## 3 Point Method Description of the axis of symmetry, finding the zeros,and a,r,s values. Vertex is AOS(x) and y intercept (subbing in aos value in the equation for x). |

## Word Problem using Factored Form

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

## Unit 3: Standard Form

## Learning Goals for Unit 3

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

## Quadratic Formula Example

## Completing the square Example

## 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

*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.*