Quadratics

Vertex Form. Factored Form. Standard Form

About

Hi my name is Nana Boahen and I am a grade 10 student . This website was created based off the grade 10 curriculum to help foster a solid foundation in quadratics for senior grades. If you have any questions feel free to contact me.

Introduction

A quadratic equation involves a term where the highest exponent is 2. The standard form of a quadratic equation is y=ax^2+bx+c where a,b and c are not 0. It can be written in vertex form and factored form.


Understanding quadratic equations allows one to apply their knowledge and solve real life problems. For example, when you throw a ball it goes up into the air, slows down and comes back down again. As a result of my understanding, I can figure out the max height of the ball.


Another place where quadratics is used Wonderland! That's right Wonderland. Engineers use their understanding of quadratics to decide the max and min height of roller coasters.

VERTEX FORM

A quadratic equation is in vertex form when it looks like this: y= a(x-h)^2+k . This form allows us to plot the vertex (h,k) on a graph and create a parabola using the step pattern.

OUR LEARNING GOALS FOR THIS UNIT ARE

1. I am able to label the parts of a parabola using terms such as vertex, optimal value, x intercept etc.


2. I am able to describe the transformations that the parabola has undergone using my understanding of the "a" value, "h" value and "k" value.


3. I am able to take my understanding of the equation: y=a(x-h)^2+k and use it to plot 3 points on a graph

SUMMARY

This was our introduction to quadratics and we learned how to determine whether an equation was quadratic or not, how to create a table of values using mapping notation, describe transformations that the graph has undergone from y=x^2 (base form) and how to apply our knowledge through solving word problems.

Finite Differences

You can tell the difference between a linear and quadratic equation using a table of values.


Linear equations have a first difference that is the same:

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Quadratic equations have a second difference that is the same:

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1. Labeling Key Features of Quadratic Relations

**The graph of a quadratic equation is referred to as a parabola (curved or "u"shaped)**


Vertex: It is the place where the graph changes direction and can either be the highest or lowest point of the parabola.

Optimal Value: The optimal value is the max or min value of the parabola in other words- the highest or lowest point. It is represented by the k value. If the parabola opens down then the optimal value is the max value and if it opens up then it is the min value.


The Axis of Symmetry: This point divides the parabola into symmetrical halves. The AOS is represented by the h value in the vertex. The equation for the axis of symmetry is x=h.


X Intercepts: The point where the parabola interacts with the x intercept and where y is 0.

2. Describe Transformations

To graph y=a(x-h)^2 + k you must first understand what each variable does. The a value effects the shape of the parabola and determines if it's stretched or compressed. If the a value is greater than 1 then it's stretched and if it's less than one then it's compressed.


y= 0.5(x-h)^2 +K (compressed)

OR

y= 5( x-h)^2 +k (stretched)


The h value determines if the parabola is horizontally translated to the left or the right.

e.g, y=a(x-5)^2+k

h=5

For example: The x value was moved 5 units to the right


*If the h value is negative it is moved to the right ( so it's positive) and if it is positive it is moved to the left (so it is negative).


The K value determines where it is vertically translated. If it is positive then it is translated up and if it is negative it is translated down.


Describe the translation:

y= 5(x-7)^2-5

The parabola is vertical stretched by a factor of 5. It is horizontally translated 7 units to the right and it is vertically translated 5 units down


or


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


The parabola is reflected down on the x axis. The parabola is vertically stretched by a factor of 2. The parabola is horizontally translated 5 units to the left and is vertically translated 3 units up.


Now you try:

y=-3(x-9)^2-7


To solidify this concept watch this video of how to graph equations in vertex form using transformations: https://www.youtube.com/watch?v=PGdP9D47XSU

3. Plotting points on a graph

y=6x^2+6

1. Determine the vertex

h=0 k=6

(0,6)

2. Plot the vertex

3. Use the step pattern (multiply the step pattern by the a factor if it's greater than 1) to make an accurate parabola.

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Video

Another way to create a graph in vertex form is by using "mapping notation"


Here is a video I created using "PowToon" to demonstrate this concept: https://www.powtoon.com/online-presentation/bZov0pd3Xcq/graphing-quadratic-equations/?mode=movie

Word Problems: Graphing Quadratic Functions Using Vertex Form

A football is kicked into the air. It's height in meters after it is kicked is given h= 4.9(t-2.4)^2+29.


a) What was the maximum height of the football and how do you know?

The maximum height is 29 cm and I know this because it is the k value which is also the optimal value.


b) What was the height of the ball when it was kicked?

t=0

h=-4.9(0-2.4)^2+29

h= -4.9(5.76)+29

h= 0.776

h=0.8

:. The height of the ball when it was kicked off the ground was 0.8 cm


c) How high was the ball after 2 seconds?

h= -4.9(t-2.4)^2+29

h=-4.9(2-2.4)^2+29

h=-4.9 (-0.4)^2+ 29

h= -4.9 (0.16)+29

h= 28.2

The ball was 28.2 cm after 2 seconds

Factored Form

Factored form is used to plot points on a graph and is an alternative to using "vertex form"


It is written like this : y=a(x-r)(x-s)

Summary

Vertex form gives us two roots and allows us to figure out the AOS and eventually the vertex so we can create a quadratic function.

Our Learning Goals For this unit are

1. I am able to use my knowledge of GCF to factor standard form equations

2.I am able to plot points on a graph using the factored form equation

3.I am able to apply my knowledge of factored form equations by solving real world problems

1.Factoring Expressions


To understand factoring you first must understand how to "expand binomials". Here is a quick video teaching you the concept: https://www.youtube.com/watch?v=JWJ-eUgP0uY



When factoring the first thing you should do is look for the GCF (flashback to grade 4 :) among the coefficients and variables.


E.g, 2h^2+6h^4

What is the GCF between 2 and 6?

The answer is 2


What is the GCF the variables have in common?

The answer is h^2


2h^2(3h^2)

This is referred to as "GREATEST COMMON FACTORING"


Aside from this rule, you can also use your previous knowledge of perfect squares when working with terms that are perfect squares.


E.g,

x^2-36

What is the square root of -36?

6 and -6!

(x+6)(x-6)


144y^2-169

What is the square root of 144 and -169?

12 and 13!

(12y+13)(12y-13)


NOW YOU TRY:


100r^2-9


The square root of 100 is 10 and the square root of 9 is -3 and 3.

(10r+3)(10r-3)


This is a special case and called "A DIFFERENCE OF SQUARES" because the second term is negative and a perfect square.


When there are 3 terms and perfect squares are involved in the equation it is referred to as a "PERFECT SQUARE TRINOMIAL"


E.g, 9x^2+24xy+16y^2

What are the square roots of 9 and 16?

3 and 4 !

(3x+4)(3x+4)

To check your answer multiple these values by 2 and make sure they equal the middle term.


Moreover, we will learn how to factor quadratic expressions of the form x^2+bx+c and this is referred to as "SIMPLE FACTORING" (when A=1).


X^2+bx+c

**There is a coefficient infront of the x^2 and it is 1.

x^2+3x+2

- find two integers that when multiplied together =2 and when added together =3

2 x 1

2+1

;. (x+2)(x+1)


or

x^2+6x+9

- what are two integers that when multiplied together =9 but when added=6

3 x 3=9

3+3=6

(x+3)(x+3)


However, questions are not always this easy because "a" doesn't always equal 1

sometimes it looks like this :


4x^2+9x+2

where a is larger than 1. These questions are referred to as "COMPLEX FACTORING"( When A is greater than 1)


There are 2 ways to go about solving these questions:


1. Guess and check

-experiment with the factors of these integers until you get the right answer


6m^2-17m+5

(6m+5)(m+1)-->not correct


(3m-1)(2m-5)--->correct

6m^2-15m-2m+5


Another method to factor "COMPLEX FACTORING" questions is by Decomposition which is demonstrated in this video: https://www.youtube.com/watch?v=1AlPfk8jx8U


Lastly the last scenario you will encounter is "FACTORING BY GROUPING". That's when there are no common variables or coefficients. This usually happens when there are 4 terms.


10x^2+5x+4x+2

- Group the first the two terms and the other two terms

-Factor out the GCF

5x(2x+1)+2(2x+1)

(5x+2)(2x+1)

2.Creating a Graph

Now that we have the knowledge we are going to learn how to create graphs using factored form.


Suppose you're given this equation:

y=x^2+2x-8

THINK: 2 integers that have a product of -8 are and equal 2

4 x -2=-8

4-2=2


(X+4)(X-2)

X+4=0 X-2=0

X=-4 X= 2


-Add the r and s value and divide by 2. The value is the AOS (the AOS divides the parabola in half)

-4+2/2=-1

-To find the y value insert -1 into both x values

y=(-1+4)(-1-2)

y=(3)(-3)

y=-9


You know have all the parts to create and label a graph!

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3. Solving word problems

Apply you're knowledge and solve these word problems:


A rectangle has area defined by 8X^2+2X-15


a) Factor to find algebraic expressions for the width and length of the rectangle


Method of choice: Guess and check

(8x-5)(x+3)

=8x^2+24x-5x-15

=8x^2+19x-15 --> wrong


(4x-5)(2x+3)

=8x^2+12x-10x-15

=8x^2+2x-15---> correct


b) If x represents 12 cm, determine the area and perimeter of the rectangle


A= LW P=2(L+W)

A=(4x-5)(2x+3)

A=(4(12)-5)(2(12)+3)

A=(48-5)(24+3)

A=(43)(27)

A=1161


P=2(4(12)-5)+(2(12)+3)

P=140


:. The area is 1161 cm^2 and the perimeter is 140 cm

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Standard Form

The standard form of the quadratic equation is ax^2+bx+c=0 where a, b and c are known values and a can't be 0.


The value of a gives you the shape and direction of the opening and the value of c is the y-intercept.

Summary

Our understanding of "standard form" ties our quadratics unit together and allows us to graph quadratic equations in any format.

Our learning goals for this unit are

1. I am able to identify the x intercepts of the equation using the quadratic formula

2. I am able to turn the standard equation into vertex form by completing the square

3. I am able to plot points on a graph using the standard form equation

1. Identifying the X intercepts of the equation using the quadratic formula

  • To identify the x intercept of the equation you have to use the quadratic formula:


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For example: Find the X intercept of this equation 4z^2-20z+25=0. You do this by inserting the values of this equation into the quadratic formula:
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Now you try solving this equation: 3q^2+19q+20

2. Turn Standard Form into Vertex Form by Completing the Square

The benefit of vertex form is that it identifies the vertex (x,y)

Rewrite each relation in the form of y=a(x-h)^2+k

This how you do it:

1.Bracket the first values

2. Divide the b value by 2 and square it

3. Include the positive and negative integer of the b value you divided and squared in the bracket

4. Put the negative value outside the bracket.

5. When applicable use your knowledge of squares from the previous unit

**If x had a negative value or a value greater then 1 you would have to multiple the b value by it**


a) y=x^2+8x+4

1. y=(x^2+8x)+4

2.y=(x^2+8x/2^2)+4

3.y=(x^2+8x+16-16)+4

y=(x^2+8x+16) -16+4

y=(x+4)(x+4)-8

y=(x+4)^2-8


For further clarification on "completing a square" please refer to this video: https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-by-completing-the-square/v/solving-quadratic-equations-by-completing-the-square


Now you try

b) y=x^2+4x+5

and

y=3x^2-12x-5~> The same rules apply but you will have to factor the a and b values by the GCF

You can know how many solutions a standard form equation has by using the formula: b^2-4ac

1. If the answer is greater than 0 then there are 2 solutions

e.g, b^2-4ac 6^2-4ac

= 36-4(2)(3)

= 36-24

= 12

2. If the answer is less than 0 then there are no solutions

e.g, b^2-4ac 2^2-4(9)(7)

=4-4(63)

= 4-252

=-248


3. If the answer is equal to 0 then there is only one solution

e.g, b^2-4ac 8^2-4(2)(8)

=64-4(16)

=64-64

=0

3. Creating a graph Using the Standard Form Equation

1. Find the X intercept

2. Find the vertex

3. Graph


1.Find the X-intercept using the Quadratic Formula



y = x ^ 2 + 2x-3 = 0

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2. Find the vertex


Get the X value and insert it into the equation to find the y value


AOS=r+s/2 and then sub into the equation


x = 1-3 / 2


x = -2/2

x = -1


y = -1 ^ 2 + 2 (-1) -3

y= 1-2-3

y = 1-5

y = -4



:. The vertex is (-1,-4)

Plot these points on a graph

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Word Problems

Ms.Boahen runs a snowboard rental business that charges 12 dollars per snowboard and averages 36 rentals per day. She discovers that for each $0.50 decrease in price, her business rents out two additional snowboards per day. At what price can Ms.Boahen maximize her revenue?


Let x represent the price reduction

R=(Profit)(Quantity)

R=(12-0.50x)(36+2x)

R=432+24x-18x-1x^2

R=-1x^2+6x+432

**Find x using the quadratic formula**

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:. She can maximize her revenue at $441

Reflection

Overall, I enjoyed this math unit because it was easy to see the connection between the various strands and where it applies in real life.

Some connections between the strands were:


1. If an equation is in standard form and you want to quickly identify the vertex then you have to complete the square -> turn it into vertex form. Sometimes you need to factor out the GCF which is only possible by having a solid understanding of factoring. This ties together your understanding of factoring ,recognizing vertex format and plotting key features of a graph such as the vertex.


2.Factored form is connected to vertex form because they both allow you to easily identify the AOS. The AOS is represented by the h value in a vertex equation and the AOS is calculated by isolating the x's and adding them together and dividing by 2 in a factored equation. This is important because the AOS divides the parabola into symmetrical halves.


3.Vertex Form is connected to graphing because you can recognize the optimal value,the vertex, the AOS, the direction, shape and transformations the parabola has undergone. This is helpful because all the key parts of a graph are identified just by looking at the equation.


4.The connection between standard form and factored form is they both help you to identify the roots.You can factor a standard equation and isolate for the x's and divide by 2 or you can solve for the x's by substituting the values from the standard equation into the quadratic formula.


5.Standard form connects to graphing because the C value represents the y intercept and "a" gives you the shape and direction .You then use the quadratic formula to find the x value and from there substitute for the y value so you know the vertex. You now have the key parts that make up a graph.


6. Vertex form connects to graphing because "a" gives you the shape and direction of the opening, the value of r and s gives you the x intercepts/AOS, you sub x into the equation to get the optimal value and set x=0 to get y. These are all the key points of a graph

This is an assessment I received back that I am proud of. This is because my ability to apply my knowledge has improved when compared to my first test and as a result I got a 92% on the Application section.
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Here are helpful websites to reference when studying grade 10 quadratics: