Quadratic Relations

By: Harman Saini

What is Parabola?

A parabola is a curved line in math, which is sometimes graphed with specific properties such as an axis of symmetry, an y-intercept, a x-intercept and a vertex.
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What are the quadratic relations?

Quadratics consists of 4 different types of relations:
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Vertex form y = a(x - h)2 + k

  • Axis of symmetry
  • Optimal Value
  • X-intercepts

What is the axis of symmetry?

The axis of symmetry is a straight invisible line, which divides the parabola in half from the center.

How do you find the axis of symmetry?

To find your Axis of Symmetry you would need to look at the h value, in the vertex form.

In the equation y= -2(x-2)2+5 the h value which is -2, turns into a positive and will become your axis of symmetry. You would write you axis of symmetry like: x= 2.

What is the optimal value?

The optimal value is the maximum or minimum, which occurs at the vertex of the parabola. If the graph opens down then the parabola has a maximum optimal value. If the graph open up then parabola has a minimum optimal value.

How do you find the optimal value?

To find the Optimal Value you need to look at the k value, in the vertex form. In this equation y= -2(x-2)2+5 the optimal value will be 5. If the optimal value is a positive number then the vertex will be over the x-axis, if the optimal value is a negative number like -5 then the vertex would be below the x-axis, and If the optimal value is 0 then the vertex will be on the x-axis.

What is the x-intercepts?

The x-intercepts are where the graph crosses the x-axis.

How to find the x-intercepts?

To find the x-intercepts in vertex form, the first step is to bring the k value to the other side of the equal sign, and change the number from positive to negative or negative to positive. After, you would find the square root of both terms on both sides by -+. Then, you would bring the number beside the x variable to the other side and would take the new equation you have found and subtract the two terms together and solve for x.


Example:


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

0=2(x+2)2+6

-6=2(x+2)2

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

-3=(x+2)2

+-3=x+2


-3=x+2

-3-2=x

-5=x


-3=x+2

-3-2=x

-1=x

How to graph a parabola in vertex form?

Graphing a parabola in vertex form

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

  • Axis of symmetry
  • Optimal Value
  • Zero/x-intercept

How to find the axis of symmetry?

To find the axis of symmetry in factored form, you must first take the numbers inside the brackets out. In this equation y=(x-4)(x-6) you must change the signs of the numbers in the bracket (-4 will become 4 and -6 will become 6). After that you must add the two number together and divide it by two, which would give your axis of symmetry as 5.

How to find the optimal value?

To find the optimal value in factored form, you must first sub in the axis of symmetry into the x variables in the brackets and then solve for y, which will give you your optimal value.


Example:


y=(x-4)(x-6)

y=(5-2)(5-6)

y=(3)(-1)

y=-3

Optimal value= -3.

How to find the zeros/x-intercepts?

To find the zero/x-intercepts in factored form, you just have to take the numbers out of the brackets, change the signs and those are your x-intercepts.

How to graph in factored form?

Graphing Parabolas in Factored Form y=a(x-r)(x-s)

Standard Form y=ax^2+bx+c

  • Axis of symmetry
  • Optimal Value

Quadratic Formula

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How to solve quadratic Formula?

Math 101 with Mr Saini

How to find the axis of Symmetry?

To find the Axis Of Symmetry using standard form, you need to get the b value of the equation and then divide it by the a value from the equation, and also change the sign of the b value according to the formula (5 turns into -5 and -8 turns into 8).

How to find the Optimal Value?

To find the optimal value using standard form, you just need to take the x value you got from the quadratic formula and replace it with the x variable in the equation.

How to solve by completing the square?

❤² How to Solve By Completing the Square (mathbff)

Discriminant

The discriminant helps you find out how many solutions the equation has. All you have to do is substitute the numbers in the equation into their variable and then solve.
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How to factor trinomials

Factoring trinomials with a common factor

How to go from Standard form to Vertex From?

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To go from Standard Form to Factored form you would put the first two terms in brackets, and then try to complete the square inside the bracket. To complete the square you would subtract the same number that you added as well, and then you take the factored out number, and use distributive property to bring that number to both terms. Then last numbers remaining would be added together or subtracted to get vertex form.

How to go from standard form to factored form?

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To go from standard form to factored form, you would have to factor out the equation. Then turn the terms in the brackets into either a simple trinomial or a complex trinomial, and you will get Factored Form.
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Quadratic Word Problems

Professor Tyagi-Quadratic word Problems
When solving a quadratic word problem you much read the question very carefully and interpret the numbers and plug them in to the quadratic equation. After that its just simply solving according the quadratic formula.

Reflection

Throughout this whole quadratic unit I learnt a lot of new methods and new ways to graph an equation, but overall I still had some difficulties understanding some of the stuff. In the quadratic mini test i did not know how to graph a parabola. Which made me lose some marks, but with critical thinking and practice I am able to understand how to solve the problems now. I also had trouble with question 7 on my test, in which i had to factor an equation, but with practice and help from the teacher i am able to understand and solve the equation. I was also able to learn many more things through this website i made. In the end Quadratics can be very easy if you work hard and put in the effort. You can also end up getting a high mark in this unit and improve your mark in the course.