Quadratics Website

By: Jay Patel

Welcome

Welcome To QuadHelp! This website will help you understand the basics of the Grade 10 Quadratics Unit, and help you get the highest marks you can. This website covers all three types of forms and gives you access to tons of Video's and examples for reference!

Table Of Contents

1. Vertex Form

- How To Find The Axis Of Symmetry

- Optimal Value

- Transformations

- X-Intercepts and Zeros

- Step Pattern

- How to Graph a parabola in Vertex Form

- How To Change Forms

- Vertex Form to Standard Form

- Vertex Form to Factored Form


2. Factored Form

- How to Find Zeros and X-Intercepts

- Axis Of Symmetry

- Optimal Value

- How to Graph a parabola in Factored Form

- How to Change Forms

- Factored Form to Vertex Form

- Factored Form to Standard Form


3. Standard Form

- Zeroes

- Axis of Symmetry

- Optimal Value

- Discriminants

- Completing the Square to go from Standard Form to Vertex Form

- Common Factoring

- Simple Trinomial Factoring

- Complex Trinomial factoring

- Perfect Squares

- Difference of Squares

- How to change Forms

- Standard Form to Factored Form

- Standard Form to Vertex Form


4. Word Problems


5. Reflection

Vertex Form ( y= a(x-h)2 + k )

Axis Of Symmetry

What Is the Axis Of Symmetry ?

The Axis of Symmetry is the invisible line that divides the parabola equally in half.

How do you find the Axis Of Symmetry?

Finding the Axis of Symmetry is kind of confusing, but if you remember the rule “Keep the number, but switch the sign”, then you will get used to how to finding the Axis of Symmetry. Now you may be wondering how to find the A.O.S. but before that, we need to know what variable represents the A.O.S. in the Equation. When looking at the equation, the variable h represents the A.O.S. Therefore the number that replaces the variable h is the A.O.S.
This may lead you to think that if the equation is y= 3(x-5)2 + 67 the A.O.S. is -5, WRONG!!!!! The thing is, in the equation there is originally a negative sign in front of the variable h, so, to find the actual A.O.S., you would take the h variable and divide it by -1. So if the equation is y= 3 (x - 7)2 + 67 then I would take the h variable, in this case -7 and divide that by -1, to get +7, and by doing that I get the A.O.S., and I would write it as x= +7. Therefore, the A.O.S. in this case is x= +7. The same way if the equation is y= 3 (x + 5)2 + 95, you would take the h variable and divide it by -1, so in this case, +5 divided by -1 equals -5, so the A.O.S. is -5.
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Optimal Value: (y = K)

To find the optimal value in the Equation y= a(x-h)2+k, all you have to do is look at the k variable and look at the sign in front of the variable. For example, if there is an equation like y= 7 (x-3)2 +4, then the optimal value would be +4, since that is the number in the place of the variable k. On the other hand if there is a question like y= 4(x-4)-4 then the optimal value would be -4.
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Transformations:

When looking at the formula of y= a(x-h)2+k, each variable shows something important to help you graph the equation.

The 'a' Variable , shows you if the direction of opening is up or down

i. A negative variable a means the parabola faces up. For example, if the a value is -3, then the parabola faces up

ii. A positive variable a means the parabola faces down. For example, if the a value is +3 then the parabola faces down

iii. If the a value is bigger than 1 (a > 1) or bigger than -1 ( a < -1), then it is called a vertical stretch

iv. If the a value is lower than one but then bigger than -1 ( -1 < a <+1) then it is called a vertical compression

The 'h' Variable, shows if the vertex moves right or left

i. A positive h value means that the vertex moves Right. For example, if the value is +8 then the vertex would more 8 spaces to the right

ii. A negative h value means that the vertex moves left. For example if the value is -8 then the vertex would move 8 spaces to the left

The 'k' Variable , shows you if the vertex moves up or down

i. A positive k, shows you how many spaces up the vertex goes. For example, if k= 6 then the vertex would move 6 spaces

ii. A negative k, shows how many spaces down the vertex goes. For example, if k=-6 then the vertex would move down 6 spaces

How To Find The Vertex

By taking the h and k value in the equation, and putting them together, in the form of co-ordinates, you would get your vertex!

Here's what questions to answer when you are asked to describe an equation

i. Find out if the parabola opens up or down

ii. Is the parabola vertically compressed or vertically stretched?

iii. How many units the vertex has gone up or down

iv. How many units the vertex has gone to the left or right

v. Find Out what the vertex is

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X-Intercepts and Zeros

In vertex form, the first step to finding x-intercepts is to bring the k value to the other side. After you would find the square root of both terms on both sides. After that, you would bring the number beside the x variable to the other side. To find the zeroes, you would take the new equation you have found and subtract the two terms together and solve for x, and at the same time, add together the numbers of the same equation and solve for x. Lastly, turn the x variables you have found into co-ordinates(by substituting the y value for 0), and you have gotten your x-intercepts!
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Step Pattern:

The step pattern is the pattern that that can be used when graphing in Vertex Form. The pattern is left or right one, up or down one, and left or right two, up or down four. This is the rule that is followed when the a value is one. When the a value is not one, then you would multiply the step pattern by that a value. For example if the a value was 2, then you would multiply the pattern by 2, so the new step pattern would be left or right one, up or down 2, left or right 2 and up or down 8.
In this example, I plotted the very simple and basic equation y=x2
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3.2 Graphing from Vertex Form

How to Graph an equation in Vertex Form

In this example I plotted the equation y=2(x-3)+4
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3.3 More Graphing from Vertex Form

How to change forms

Vertex Form to Standard Form

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To go from Vertex Form to Standard Form you would have to multiply the brackets together and then multiply the a value with everything that's inside the bracket, and then solve!

Vertex Form to Factored Form

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To go from Vertex Form to Factored Form, you would have to isolate x, and then find the x-intercepts and zeroes, and then you would take the x-intercepts and the original a value, put it together and there you have it, Factored Form.

Factored Form ( y= a(x-r)(x-s) )

Zeros Or X- Intercepts ( r and s)

When looking at the factored form, the two brackets are actually the two x-intercepts of the parabola. Finding the two intercepts of the parabola is quite easy! First, you take the terms in both brackets and make them equal to zero. Then, you would isolate for x, and you will get your two x-intercepts! After that you would take your x-intercepts and turn then into co-ordinates, by making the x value the x-intercept, and making the y value 0.

Axis Of Symmetry: ( x= (r+s)/2)

The Axis of Symmetry is the invisible line that divides the parabola equally in half, and the Axis of Symmetry helps in finding the optimal value. To find the Axis of Symmetry you would add the two x-intercepts you found when finding x-intercepts and then divide the answer you get by 2 to get the A.O.S.

Optimal Value (Sub In)

To find the Optimal Value, you would take the Axis of Symmetry you just found and replace it with the x variable in the equation. After that by getting the “y” value you just found, you would take that number and your A.O.S. that you found and get your vertex!

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3.13 Finding Vertex from Factored Form

How To Graph using the Factored Form

When using Factored Form to graph, it is very easy to draw your parabola. First, plot the two co-ordinates you found when finding the x-intercepts. Then make your axis of symmetry and then plot your vertex. And lastly, connect the dots in a curve, and your done!
In this example I plotted the equation: y= 4(x-3)(x-5)
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3.5 Graphing from Factored Form

How to change forms

Factored Form To Vertex Form

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To go from Factored form to Vertex Form, you would have to find the x-intercepts and zeroes and then with those values, find the axis of symmetry. After finding the axis of symmetry, you would have to replace x with the A.O.S. and find the optimal value. You would place the A.O.S. in place of the h variable, and place the Optimal Value in the K variable's spot, and then you will have gotten your equation!

Factored Form to Standard Form

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In this case, all you would have to do is to expand and simplify, and there you have it, you have quickly and easily gone from Factored Form to Standard Form

Standard Form ( y= ax2 + bx + c )

Quadratic Formula

The Quadratic Formula is used to find the x variable in Standard Form. In this, the terms in the Standard Form Equation are substituted into the Quadratic Formula, according to what variable they are in the Equation, meaning, the a value in the equation will replace the a value in the formula, and the same goes for the b and c values. After doing the Quadratic Formula, you will be able to find what x equals and then find the optimal value, by substituting the variable x in the equation with the new x value you have found.
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3 15 Quadratic Formula

Axis Of Symmetry

To find the Axis Of Symmetry, all you need to do is get the b value of the equation and then divide it by the a value from the equation times two. Since the b value in the formula is a negative, when you are replacing the b value, remember to change the sign. So, 4 turns into -4 and -5 turns into 5.
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Optimal Value

Like how you did in Factored Form, you would take the x value you got from the quadratic formula and then replace it with the variable x in the equation. Then you would get your y value and with your A.O.S., you found the vertex of the parabola of the equation.

Examples Of Finding the Zeroes, Axis of Symmetry and Optimal Value

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3.14 Completing the square

Discriminants

The discriminant is +b^2-4ac, and this expression will actually help you find out how many solutions the equation you have has. All you have to do is substitute the numbers in the equation into the appropriate variable spots and then solve!
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Types Of Factoring

Common Factoring

When doing common factoring, you would look at the GCF, and divide the equation by the GCF. You would then put the equation after the division in brackets, and then the GCF outside the bracket.
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Simple Trinomial Factoring

A simple trinomial is when the “a” value is positive 1 in the equation. To factor a simple trinomial, you would look at the ‘b’ value and the ‘c’ and try to find two numbers that add up to the ‘b’ value and multiply to get the ‘c’ value. Then by creating 4 brackets, you would write x and then the first number that you got when adding and multiplying in the first two brackets and then write x and the second number in the second bracket.
3.8 Factoring Simple Trinomials

Examples

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Complex Trinomial Factoring

Before starting the actual method of Complex Trinomial Factoring, you would try to common factor the expression you have. After that, you can start the method of Complex Trinomial Factoring. To use this method, the a value can be any number that's not 1 or -1. To factor a Complex Trinomial, you would multiply the a value and the c value to get a product, then look at the b value. After, you would find two numbers that can be multiplied to get the product you got, and those same two numbers should add up to get the b value. Next, substitute the b value with the two numbers you found, and then common factor the first two numbers in one bracket, and then the last two numbers in another bracket, and then you have Factored a Perfect Square Trinomial!
3.9 Complex Trinomial Factoring

Examples

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Perfect Squares Factoring

How do you know if a number can be turned into a perfect square? If a number has a whole number as a square root, then that number can be turned into a perfect square. For example, 25 is a perfect square because 25= 5^2. When looking at an equation, if the first term and last term are perfect squares, then the equation has to be factored by the Perfect Square Factoring Method. So, first you would find the square roots of the first and last terms. After, the two square roots you found, you would take them and multiply both of them together and then multiply that by 2. If the answer you got matches the middle term in the original equation you got then you are doing the steps right, but if you don't, then you have to erase everything and then factor the equation with Complex Trinomial Factoring. After, you would take the two square roots you got and put them together in a bracket, and then outside of the bracket you would put a 2 (Squared), and your done!
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Factoring perfect square trinomials
Perfect Squares (Dark Horse Parody, Katy Perry)

Difference Of Squares

To find the difference of squares, you would first find the square roots of both terms. After that, you would make two pairs of brackets. In the first pair, you would write your a term plus your b term. In the second bracket, you would put your first term minus your second term. This is because when you are to expand the brackets, in your final answer, the first term should be subtracting the second term, and to have that as a final answer, your brackets should have a different sign.
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Factoring difference of squares

How to change forms

Standard Form to Factored Form

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In this case, first you would have to common factor the equation. After that, the terms in the brackets turn into either a simple trinomial or a complex trinomial, and you would be able to factor the trinomial out as well, and by doing that you have successfully gone from Standard Form to Factored Form

Standard form To Vertex Form

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First you would group the first two terms in a pair of brackets, and common factor, but make sure the co-efficient of both terms are being simplified, nothing else. After that, you would try to complete the square inside the bracket, and since you can't randomly add numbers in an equation, you would subtract the same number that you added as well. After, you would notice that there is a perfect square trinomial in the brackets and you can turn it into a perfect square. After there is still a number left, so then you would take the number you common factored out in the beginning, and use distributive property to bring that number to both terms. Since now ,the last numbers remaining would be added together or subtracted to get vertex form.

Word Problems

Example #1

A soccer ball is kicked into the air and follows a parabolic path described by the equation, h=-2(t-4)2 +15.

a. When does the soccer ball reach its maximum height?

b. What is the maximum height reached?

c. What is the height of the soccer ball at 6 seconds?

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For 1a) All I had to do was look at the x value of the vertex, in the equation, because that showed me how long it took to reach the maximum height


For 1b) All I had to do was look at the k value because that showed me the y value of the vertex, which shows how big the height is


For 1c) All I had to do was replace t with 6 and then solve


Example #2

The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h=-5t2+20t+60, where “t” is the time in seconds, and “h” is the height, in meters.

a. Write the above formula in factored form

b. When will the rock hit the water?

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For 2a) I went from Standard from to factored Form


For 2b) I found the x-intercepts

Example #3

A concert promoter models the profit from her next concert, P dollars, by the equation P=-11(t-55)2+10571, where t dollars is the cost per ticket.

a. What price should she charge to maximize her profit? What is the maximum profit?

b. What would be her profit if the tickets were free?

c. How much should she charge per ticket to break-even?

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For 3a) Since i know what the vertex was for this equation, I knew the ticket price would be the x value and the maximum profit would have been the y value


For 3b) To do this, you would replace the variable t with 0 and then solve


For 3c) You would change p to 0 and then solve.

Example #4

A flare is released into the air following the path, h=-5(t-6)2+182, where h is the height in meters and t is the time in seconds.

a. What was the flare’s maximum height?

b. What was the flare’s initial height?

c. How long was the flare in the air?

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For 4a) I looked at the k value to get my maximum height because the k value is also the y value of this equations vertex


For 4b) I put the variable t equal to 0 and then solved it


For 4c) I put the height to zero, since the height would be zero when the rocket hits the ground and then solved

Example #5

The length of a rectangle is 5 cm greater than its width. The area is 104cm2. Find it's dimensions of the rectangle, to the nearest hundredth of a meter.
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First, I did my let statements and then substituted my values into the equation A=l x w, and then, by using the rainbow method, and then bringing the 104 to the other side, made a simple trinomial. After that I found the zeroes and x-intercepts, and since there was one negative answer and one positive, I picked the positive number because in this case, x can not be a negative

Example #6

The CSS SAC is planning a dance, Ms. Semler has asked them to make sure they maximize the income from dance tickets sold.

Ms. Semler has told them:

- At a cost of $3 per ticket, there would be 700 students at the dance

- If the prices will go up, less students will come

- Every time the ticket price goes up by $1, the will lose 70 students.


What price should SAC charge to maximize revenue?

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By first doing the let statements, I was able to answer this question. In the equation, the first pair of brackets shows the amount of students coming, and the second pair of brackets shows the price. I then made my equation into standard form,but then had to go back to factored form to get the x-intercepts and zeroes. This helped me find the A.O.S. which in turn helped me find the optimal value. This helped me reach my conclusion that the maximum revenue will be $5200.83 with a ticket of $10.16

Reflection

I chose to showcase my lowest mark in this unit, because of the amount of lessons that I learnt from it. We are all human, we are all prone to mistakes, but it is up to us to look at what we did wrong and see how we can not make the same mistake again. The same way, the test that I am showing you right now has tons of mistakes in it, but only after clarification from my teachers and friends have I understood my mistakes. After thoroughly going through the questions and reflecting on what I did wrong, my marks slowly went up, and everything made sense to me. Surprisingly, in a personal level, this unit has actually helped me get closer to my grade 11 friends. Last year whenever they needed help in any subject, even if I didn't understand it, I would try to clarify things and help them out. Due to my large extra-curricular activity schedule, meeting up with my friends isn't as easy as it used to be, and I was scared to ask for help, but in the end they did help me. Because of their help, I am now able to fully and clearly understand the third mini unit we are focusing on. At an academic level, this unit has helped me in ways that I will only notice in the future. I know that this unit will be more talked about on a deeper level in Grade 11, and by having a thorough understanding of this unit, I would be able to not only get good marks in that unit, but to understand the whole course in general. In conclusion, even though personally I didn't like making this website and I think there would be a better way to show you teachers that we understand this unit, I have learned lots in this unit which I will carry on into next year, and the years to come.