# Quadratic Relationships

### Design to teaches students about the Quadratic Relationships

## Contents

- What's a Parabola?
- What can you figure out by using Parabola?
- Types of Quadratic Equation

2. Terminology

3. Diagram of the terms

4. Vertex Form

- Transformations
- How to create equations in vertex form?
- How to graph equations in vertex form?
- Tips
- Word Problem

- How to solve equations in factored form?
- Graphing equations in factored form
- How to determine an equation from a graph?
- Factoring Trinomials
- Graphing Trinomials
- Word Problem

- Quadratic formula
- How to solve equations by converting standard form to factored form?
- How to graph quadratic equations in standard form?

7. Converting equations to

- Standard form
- Factored Form
- Vertex form

8. Perfect square and difference of square

9. Quiz

10. My Reflection

- Assessment
- Reflection

11. Other websites to check out!!

## Introduction

## What's a Parabola?

**Parabola**is an u-shaped curve that is created by using vertex, the y-intercepts, x-intercepts and other coordinates on a graph. The parabola also has a mid line called the Axis of symmetry (AOS) and the optimal value (minimum or maximum point of the parabola) that help to identify the vertex. To sketch the parabolas, first we have to figure out these key points. This is done by using the equations.

## What can you figure out by using Parabola? Why is it important?

## Types of Quadratic Equations?

## How to identify a quadratic relationship?

## Terminology

## 1. Vertex

## 2. Axis of Symmetry

For example: The x-intercept are (6,0) and (2,0). 6+2=8. 8 divided by 2 equals 4. The axis of symmetry is 4.

## 3. Optimal Value

Example:

## 4. X-Intercept

## 5. Y-Intercept

Example:

## Diagram of the Terms

## Vertex Form

*y*=*a*(*x*–*h*)^2+*k.*## Transformations

How does "a" effect the orientation and shape?

- If a > -1, the parabola opens
**up** - If a < 1, the parabola opens
**down** - If -1 < a < 1, the parabola is vertically
**stretched** - If a >1 or a < -1, the parabola is vertically
**compressed**

How does "k" effect vertex?

- If k > 0 then the vertex moves
**up**by k units - If k < 0 then the vertex moves
**down**by k units

How does "h" effect the movement of the vertex?

- If h > 0 then the vertex will move to the
**left**side - If h < 0 then the vertex will move to the
**right**side

## How to create equations in vertex form?

Steps to solve the question:

- Plug in the given information in the formula [ y = a(x - h)^2+K]
- The
**x**value in the vertex isvalue in the formula.*h* - The
**y**value in the vertex isvalue in the formula.*k* - To complete the formula you have to solve for
*a* - Plug in the intercepts given and replace them with
and*y*.**x** - Check your work
- At the end when you figure out the value of
*a, rewrite*the formula that includes,**a**and**h**values.**k**

*Use the steps to solve the problem below.

Information given:

- Vertex: (8,2)
- Goes through the point: (6,-14)

## How to graph equations in vertex form?

## Tips

- Remember to always check your work whether if it makes sense or not.
- Be careful of determining the positive and negative signs.
- Remember the formula

## Word Problem

**A badminton player hits the birdie which follows a parabolic path. It is represented as a equation**

*h = -1(t - 3)^2 +11.*Where*h*is the height in meters, and*t*is time in the air in seconds.**When does the birdie reach its maximum height?**

A) 8 seconds

B) 3 seconds

C) -3 seconds

D) -4 seconds

The correct answer is **B (3 second).** The vertex is the highest point in this graph and 3 seconds will be the *t* value for the vertex. This is because h value ( where 3 is) is the x coordinates of the vertex.

**What is the maximum height reached of the birdie?**

A) 11 meters

B) 10 meters

C) 4 meters

D) -4 meters

The correct answer **A (11 meter).** A easy trick to solve this type of question is that when ever it says maximum or minimum, they are referring to the optimal value and the optimal value is in the vertex. We know that *K=Y*, in this case the* y *value is replace with h but it is the same thing. So therefore we look in the formula and see what the *k* value and its 11 meters.

**What was the height of the birdie when its released?**

A) 0 meter

B) 5 meters

C) 1 meter

D) 2 meters

The correct answer is** D (2 meters).** In order to identify the answer first you have to make the* t *value to 0 (zero). This is because when the birdie is released, there has not been any travel time for the birdie. Then your solve the equation to get the value of h.

## Factored Form

**Definition:** An algebraic form in which none of the expression can be simpler by putting a common factor because it factored to the max.

**Equation:**The equation used for factored form is

**y=a(x-r)(x-s)**## How to solve equations in factored form?

*X-Intercept/Zeros/Roots*For the factored form, we have sub in 0 as y to find the x-intercepts. When the parabola touches the x axis, it will have 0 y coordinates so therefore you can sub it in as 0 (zero). Once that done then you take both the pairs individually [(x-r) and (x-s)] and make them equal to 0 (zero). Then you solve for the value of *x.*

*Lets try it

y=3(x-5)(x-10)

*Axis of symmetry*Like I mentioned before, axis of symmetry is the middle x line of a parabola. Also, its the x value in the vertex. To find the axis of symmetry you have add up the x-intercepts and divide by two. [ x = (r+s)/2]

x = (5+10)/2

x = 15/2

x = 7.5

Therefore our axis of symmetry for this equation is 7.5.

*Optimal Value*In order to find the optimal value, you must first find the axis of symmetry. Axis of symmetry and optimal value are interlinked. The optimal value is the y value in a vertex. We already know the x value ( axis of symmetry) in the vertex. Now all you need to do is sub in the x value (axis of symmetry) to find the optimal value.

y = 3(x-5)(x-10)

y = 3(7.5-5)(7.5-10)

y = 3(2.5)(-2.5)

y = 3(-6.25)

y = - 18.75

Therefore our optimal value is -18.75

*Vertex*The vertex is made up of optimal value and axis of symmetry. The optimal value for this equation is -18.75 and the axis of symmetry is 7.5. Therefore the vertex is (7.5, -18.75)

## Graphing equations in Factored form

**For information on how to do graphing factored form, please watch this video.**## How to determine an equation from a graph?

## Factoring Trinomials

*Common factoring*The first thing you should * always* do while factoring equations in standard form to factored form is common factoring. You should look to find a similar component that each term can divide into. The common factor is not always a numerical number only, sometimes it can be a variable ( for example x or x^2) with a coefficient. Some equations might not have any common factors then you should leave as it is and do the problem.

Example of Common factor:

1. In order to wirte the common factor, you must first find the number that can be divided by all the numbers.

4x^2+ 8x

4x can be divided by all the numbers

2. Once you found the number by dividing it, then you rewrite the factored equation with the number that you divided being first followed by brackets and the left over equation that can not be divided any further.

4x(x^2+2x)

Example of a non-common factor:

x^2+21x-6

*Simple Trinomials*

*Complex Trinomials*## Graphing Trinomials

X-Intercept

In order to graph a trinomial equation, u have to first find the x-intercepts. First of all, you have factor the equation ( look above to see how to factor trinomials). Then just like vertex form, you have to make the y value equal to 0 (zero) to find the roots.

Vertex

Also, in factored form, you can find the vertex by solving for axis of symmetry and optimal value. The axis of symmetry is solved by adding the two x-intercept and dividing by two. To find the optimal value, you have plug the axis of symmetry in the factored equation. This will give you the vertex.

Y-Intercept (optional)

If you want to go one step further, then you can solve for the y-intercept. In order to find the y-intercept, you have to let x=0 (zero).

## Word Problem

Example:

Graph y = 2x^2 - 24x + 20 using the x-intercept and vertex. If needed, also you the y-intercept

Solution:

## Standard Form

**y= ax^2 + bx + c****How solve **

Two ways to solve:

- Quadratic formula
- Changing standard form into factored form

## Quadratic Formula

**What is the Quadratic formula?**

The coolest thing about the Quadratic formula is that it will always work. Sometimes, when using factored form, you might not be able to get the answer because it does not work with it. But, in the Quadratic formula, the equation will always works does not matter is the equation can be factored or not.

- There are 3 coefficients in the Quadratic formula. They are
*a, b*and*c,*and in the equation they are replaced with numerical numbers to help solve the equation. - Most often, Quadratic formula is used when the standard form is unable to be converted into factored form.
- The equation is Quadratic formula must equal to 0 (zero)
- Discriminant is the part in the Quadratic formula after the square root sign. It is b^2-4ac.
- If discriminant is more than 0 then there are 2 x intercepts. If its less then 0 then there are 0 (zero) x-intercept. And if the discriminant is 0 then there is 1 x-intercept.

Fun tool to learn the Quadratic Formula

*How use the Quadratic Formula?*## How solve equations by converting standard form to Factored form?

## How to graph quadratic equations in standard form?

## Word Problem

## Converting Equations

## Vertex form

**Converting from Factored form to vertex form**- Find axis of symmetry by adding the two x-intercepts and dividing by two.
- Sub in the axis of symmetry
**(h value**) to get optimal value (**k value**). - Sub in one of the point to get the
**a value**.

**Converting from Standard form to vertex form.**

- Completing the squares

## Standard Form

*Converting from vertex form to standard form*- Expand and simplify

Example:

y=2(x+3)^2-2

=2(x+3)(x+3)-2

=(2x+6)(x+3)-2

=2x^2+6x+6x+18-2

=2x^2+12x+16

**Converting from factored form to standard form**

- Expand and simplify

Example:

y=-3(x+1)(x+5)

=-3(x^2+5x+1x+5)

=-3(x^2+6x+5)

= -3x^2-18x-15

## Factored form

*Converting from vertex form to factored form*- Make
equal 0 (zero) and find the x-intercepts.**y** - Remember when you square roots a number, it will be positive and negative.
- Once you found the x-intercepts then place them into a factored form equation with keeping the "a" value as the same and removing the "k" value.

Example:

**Converting from standard form to factored form**- Factor the equation using common factoring, grouping, difference of squares, perfect square, simple and complex trinomial. ( Go above to see more examples on how to use it)

Example:

## Prefect square and Difference of squares

**Difference of squares**

*Perfect Square*## Quiz

**Please go on the link below to test yourself on everything you learned on this website. Also when doing equations and problems please remember to read the question properly and understand what its asking for before answering. Remember to check your work and be careful of using appropriate sign ( positive and negative signs) **

## My Reflection

## Assessment

- I clearly understood the different types of quadratic equations and I knew how to use it properly.
- I did well on the word problem questions because I knew what they were asking for.
- I had a good understanding of factoring equations from standard form. I had no mistakes in that.

Things I can improve on:

- I did not check my work properly so therefore I lost a lot of marks by putting a positive sign where there should be a negative sign. Next time I should check all my work before handing in the test.
- I messed up on identifying the difference between difference of square and perfect square. I should revise the lesson about difference of square and perfect square. This will prevent me from making mistakes.
- I need to completely simplify when asked in a question. I lost few marks because of this.

## Reflection

Throughout this unit, I was always able to do well on questions in class and other activities we did in class, but I was never able to get the marks I wanted on test and quizzes. I was not doing completely horrible, but I except ** way more** from myself. I was expecting to get around a 95% at the end of this unit, but right now I only have 87%. I think this is because of the lack of momentum I had on studying. In the beginning part of this unit, I felt this is too easy so I did not study that hard for the test. Also I did check my work properly during the test. I lost around 5 to 8 marks this unit, just because I did not check my work properly. But, now I have learned to check my work and this learning experience will really benefit me in the future.

All in all, for me this was not the best unit in terms of the marks I got, but It was a great learning experience. I learned essential math concepts that are necessary for me in the future and also I learned to check my work over and over again before handing it in. This key things that I have attain from this unit will really help me succeed in the future. I think that the only reason I was able to understand this unit so well is because of my teacher, Ms.Johal. She help me whenever I need it and also told me where I was going wrong. Her support and teaching was the only reason I liked this unit. Thanks Ms. Johal, you are great!!