Quadratic Relationships

Sarah Qarizadha

Welcome Everyone!

Hello everybody, my name is Sarah Qarizadha and I will be teaching you a little about the grade 10 curriculum. Today i'll be teaching you about Quadratic Relations. I will be teaching you how to answer Vertex Form, Factored Form and Standard Form questions. All of this is apart of the grade 10 curriculum, so if your brain is looking for a good lesson or just a refresh, then this is a great website for you!

Learning Goals

  1. To be able to solve all vertex form equations
  2. Finding the value of 'a'
  3. To use quadratic equations to solve word problems in vertex form

Introduction

Definitions Of Unit #1

Parabola : This is a curve, shaped as a arch. Its distance from a fixed point is equal to its distance from a fixed line

Vertex Form : The vertex are the points (h,k) on a parabola. The vertex is when both the points meet. The x-intercept is 'k' and the y-intercept is 'h'.

AOS (Axis Of Symmetry) : The AOS is a straight vertical line that goes right down the middle of the parabola

Vertex Form

y = a (x-h)^2 + k

Introduction To Vertex form

  1. Vertex is (h,k)
  2. 'a' tells us if its stretched or compressed, and the direction of opening
  3. 'h' tells us if its going to be left or right on the graph/horizontal translation
  4. 'k' tells is if its a vertical translation
  5. Use the step pattern
  6. To find the y-intercept, set x=0 and then solve for the y-intercept
  7. Then to solve, set y=0 and solve for x or expand and simplify to get the standard form
  8. Then use the Quadratic Formula

Equation Question Examples

Example #1 - Determine an equation when given the vertex and determining the 'a' value:


Vertex : (-3,5)

One of the points are (3,7)

Step 1 : Plug in the vertex into the equation

y=a(x-h)^2+k

y=a(x+3)^2+5

Step 2 : Find the 'a' value. Plug in (3,7) in the equation

y=a(x+3)^2+5

7=a(3+3)^2+5

7=a(36)+5

7=36a+5

7-5=36a

2/36 = 36a/36

18=a

Step 3 : Fill in final equation

y=18(x+3)^2+5

Example #2 - Determining an equation when given the vertex and when given (x,y) :

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Example #3 - Isolating For 'x' :

*We are finding the x-intercept when we are isolating for 'x'

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Graphing Vertex Form


We can easily use a step pattern to graph y = 2(x-4)^2 - 6

In the step pattern, your supposed to multiply the 'a' value by 1, 3 and 5

Vertex : (4,-6)

AOS : -4

Optimal value : -6

'a' : 2

Direction Of Opening : Up, because the 'a' value is positive


By using the step pattern, we can easily figure out what parabola will be :



  • Step 1 : 1x2 = 2
  • Step 2 : 3x2 = 6
  • Step 3 : 5x2 =10
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Word Problem

Using Vertex Form


At a baseball game, a fan throws a baseball from the stadium back onto the field. The height in meters of a ball t seconds after being thrown is modeled by the function h = -4.9 (t-2)^2 + 45


a) What is the maximum height of the ball?

The Vertex : (2,45)

Maximum height of the ball is 45 meters

b) When did the maximum height occur?

The maximum height is 2 seconds

c) What is the height of the ball after 1 second?

h = -4.9 (1-2)^2 + 45

h = 40.1 meters

Therefore, the height of the ball is 40.1 meters after 1 second

d) What is the initial height of the ball?

When t=0 Solve for h

h = -4.9 (0-2)^2 + 45

=25.4

Therefore, the initial height was 25.4 meters

First and Second Differences

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Mapping notation

(x,y) - (x+h, ay+k)

Example - y=(x+7)^2 ~ (x-7, y)

Graphing Using Transformations

Quick Way of Graphing a Quadratic Function in Vertex Form

Factored Form

Learning Goals

  1. To be able to graph a quadratic equation in factored form
  2. Turning Factored Form, Quadratic Equations into Standard form by using FOIL and simplifying

Factored Form Equation

y = a (x-r) (x-s)
Factors And Zeros


A zero of a parabola is another name for the x- intercepts and in order to find the x- intercepts you must set y=0


Key points

  • When the (a) value changes the zeroes do not change
  • When the (a) value changes the axis of symmetry do not change
  • When the (a) value changes the optimal value does change

Introduction To Factored Form


The value of a gives you the shape and direction of opening
The value of r and s give you the x-intercepts
axis of symmetry, AOS: = (r+s / 2)


  1. Sub this x value into equation to find
  2. the optimal value
  3. to find the y-intercept, set x=0 and solve for y
  4. Solve using the factors

Types of Factoring:
  • Greatest Common Factor
  • Simple factoring (a=1)
  • Complex factoring
  • Special case - Difference of squares
  • Special case – Perfect square

Equation Question Examples

Factoring Simple Trinomials


  1. Identify a,b, and c in the trinomial ax^2 + bx + c
  2. Write down all factor pairs of c
  3. Identify which factor pair from the previous step sums up to b
  4. Substitute factor pairs into two binomials

Factoring Complex Trinomials

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Factoring by grouping

Step 1: Decide if the four terms have a GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.


Step 2: Group first two terms together and the last two terms together.


Step 3: Factor out the GCF from each of the two groups.


Step 4: The one thing that the two groups have in common should be what is in parenthesis, write whats outside the brackets as a parenthesis


Step 5: Determine if the remaining factors can be factored any further.

Graphing Factored Form

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Common Factoring

Step 1: Determine the GCF of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common.


Step 2: Factor out or divide out the GCF from each term. You could check your answer at the point by distributing the GCF to see if you get the original question. Factoring out the GCF is the first step in many factoring problems.

Word Problem Using Factored Form

A toy rocket is shot into the air. It's path is approximated by the function h = -5t ^2 + 25t where h(t) is the height in meters and t is the time in seconds. When will the rocket hit the ground?


h = -5t ^2 + 25t

h = -5t (t-5)

t=0 t=5

Therefore, the rocket hit the ground at 5 seconds

Special Cases

Difference Of Squares

Equation

a^2 - b^2 = (a - b)(a + b)

or

a^2 - b^2 = (a + b)(a - b)



Step 1: Decide if theirs a GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.


Step 2: So, all you need to do to factor these types of problems is to determine what numbers squares will produce the desired results.


Step 3: Determine if the remaining factors can be factored any further.

Perfect Squares

Equation

(a + b)^2 = a^2 + 2ab + b^2


(a - b)^2 = a^2 - 2ab + b^2



Step 1. Verify that the first term and the third term are both perfect squares. (This means that the coefficients are perfect squares


Step 2. Verify that the middle term is twice the product of the square roots of the first and third term.


Step. 3. Use the standard form above to write the factored form.

Factored Form - MY VIDEO

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Standard Form

learning goals

  1. Find the number of zeros that a quadratic relationship has by calculating the discriminant.
  2. Solve using the quadratic formula or by factoring or by completing the square to get vertex form

Standard form equation

y = ax^2 + bx + c

Introduction to standard form


The value of a gives you the shape and direction of opening

The value of c is the y-intercept

Solve using the quadratic formula, to get the x-intercepts

MAX or MIN? Complete the square to get vertex form

Quadratic equation : The quadratic equation is a formula used to get the x-intercepts. I recommend using this, if the equation isn't easily factor-able.

Quadratic FORMULA example

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The Discriminant : A discriminant is to know how many x-intercepts there will be. Using this formula: b^2 - 4ac


  • There can be two solutions, one solutions or no solution.
  • There are two solutions when the discriminant is a positive.
  • There is one solution when the discriminant equals to 0
  • There is no solution, when the discriminant is negative
Turning Into Vertex Form And Completing The Square :


Question: y=3x^2 +6x -4


  • Step One: Focus on the 3x^2 +6x, and factor out the GCF (which is 3 in this case) and leave the -4 out.

y=3(x^2 +2x) -4

  • Step Two: Divide the 'b' value by 2 and then square the result.
(2/2)^2=1
  • Step Three: Add the result in the bracket and also subtract it. So the equation looks like this: y= 3(x^2 +2x +__ -__) -4
y=3(x^2 +2x +1 -1) -4
  • Step Four: Take the negative one outside of the brackets, therefore it will be multiply with 3.
y=3(x^2 +2x +1) -3 -4
  • Step Five: factor the trinomial in the bracket.
y=3(x+1)^2 -7
  • *Remember Vertex Form Equation* y=a(x-h)^2+k
  • Therefore, vertex is (-1, -7)

word problem using quadratic formula

1) A toy rocket is launched upward at an initial velocity of 51 m/s, from a height of 1.3 , above the ground, the height of the toy rocket, in meters, after t seconds is modeled by the equation h= -4.9t^2 + 51t +1.3


a) How long does it take the rocket to fall to the ground, rounded to the nearest hundredth of a second?


b) Find the times when the toy rocket is at a height of 95.7 m above the ground. Round your answers to the nearest tenth.


c) What is the maximum height of the toy rocket? At what times does it teach this height? Round you answer to the nearest tenth.

Video explaining standard form - Quadratics

Using the Quadratic Formula

REFLection on quadratics

Quadratics was a new unit for me where I got to learn many interesting things about parabolas. I first struggled with parabolas, but later on, I understood it more and more. I feel like if I redid my three unit tests for quadratics, then I would do fairly well. Making this website has made me understand the concepts of quadratics much more. Below is an image of my Quadratics Standard Form Unit Test. I chose to show this test, because i feel that this unit was my favorite. It was an easy unit to understand, and i enjoyed answering the questions for the different quizzes and tests. I chose to not show any of the tips assignments because I did not get the marks that i thought i would get. I feel like if got to redo any assignment or test, then it would be the various Tips assignments. Overall, i know that if i were to ask more questions in class, finish my homework and practice with extra worksheets, then i would be passing math with over an 85% for sure.
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Connections between quadratic relations


All the three forms of equations are capable of being graphed. From each type pf equation you will need to find the x- intercepts also known as the zeroes then from the equation you can find the y and x value of the vertex. These enable you to plot three points so you can graph. An example of a connection is for standard form, when you complete the square you still have to rearrange it into vertex form. then you'll get the vertex (h and k). Another connection i made between units, was the graphing. when we graph for any unit, you can always find the axis of symmetry (AOS). By graphing, you can also find the x-intercepts/zeros. By completing either Vertex, Standard or Factored form, you can find the vertex for any of the equations and then graph it, which is also a similar connection between all the units. These are some connections you could make between all three units.